4.15 Application: Gaussian Elimination
223
V (Di ) = {vπ(ni +1) , vπ(ni +2) , . . . , vπ(ni+1 ) }.
It is easy to see that B has (n1 , . . . , np )-block-triangular structure. This implies that A has block-triangular structure. The above observation suggests
the following procedure to reveal hidden block-triangular structure of A.
1. Replace every non-zero entry of A by 1 to obtain a (0, 1)-matrix B.
2. Construct a directed pseudograph D with vertex set {v1 , . . . , vn } such
that B is the adjacency matrix of D.
3. Find the strong components of D. If D is strong, then B (and thus A)
does not have hidden block-triangular structure6 . If D is not strong, let
D1 , . . . , Dp be the strong components of D (in acyclic order). Find a
permutation π on {1, . . . , n} such that
V (Di ) = {vπ(ni +1) , vπ(ni +2) , . . . , vπ(ni+1 ) }.
This permutation reveals hidden block-triangular structure of B (and
thus A). Use π to permute rows and columns of A and coordinates of x
and b.
To perform Step 3 one may use Tarjan’s algorithm in Section 4.4.
We will illustrate the procedure above by the following example. Suppose
we wish to solve the system:
x1
+ 3x3
x2
2x1 + 2x2 + 4x3
3x2
+
+
+
+
8x4
5x4
9x4
2x4
= 2,
= 1,
= 6,
= 3.
We first construct the matrix B and the directed pseudograph D. We
have V (D) = {v1 , v2 , v2 , v4 } and
A(D) = {v1 v3 , v1 v4 , v2 v4 , v3 v1 , v3 v2 , v3 v4 , v4 v2 } ∪ {vi vi : i = 1, 2, 3, 4}.
The digraph D has strong components D(1) and D(2) , which are subdigraphs
of D induced by {v1 , v3 } and {v2 , v4 }, respectively. These components suggest
the following permutation π, π(i) = i for i = 1, 4, π(2) = 3 and π(3) = 2, of
rows and columns of A as well as elements of x and b, the right-hand side.
As a result, we obtain the following:
+ 8x04 = 2,
x01 + 3x02
0
0
0
2x1 + 4x2 + 2x3 9x04 = 6,
x03 + 5x04 = 1,
3x03 + 2x04 = 3,
6
Provided we do not change the set of entries of the diagonal of A
224
4. Classes of Digraphs
where x0i = xi for i = 1, 4, x02 = x3 and x03 = x2 .
Solving the last two equations separately, we obtain x03 = 1, x04 = 0.
Now solving the first two equations, we see that x01 = 2, x02 = 0. Hence,
x1 = 2, x2 = 1, x3 = x4 = 0.
A discussion on practical experience with revealing and exploiting blocktriangular structures is given in [208].
4.16 Exercises
4.1. Let φ(u) be the forefather of a vertex u as defined in Section 4.4. Combining
(4.2) and (4.3), prove that φ(φ(u)) = φ(u).
4.2. Prove Proposition 4.3.1.
4.3. Prove Lemma 4.4.1.
4.4. In part (ii) ⇒ (i) of Theorem 4.5.1, prove that σ(D) = L(Q).
4.5. Derive Corollary 4.5.2 from Theorem 4.5.1 (iii).
4.6. (−) Prove Proposition 4.5.3 using Theorem 4.5.1 (i) and (ii).
4.7. Prove the following simple properties of line digraphs:
~n−1 if and only if D ∼
~n ;
(i) L(D) ∼
=P
=P
∼
~
~
(ii) L(D) ∼
C
if
and
only
if
D
C
= n
= n.
4.8. Let D be a digraph. Show by induction that Lk (D) is isomorphic to the
digraph H, whose vertex set consists of walks of D of length k and a vertex
v0 v1 . . . vk dominates the vertex v1 v2 . . . vk vk+1 for every vk+1 ∈ V (D) such
that vk vk+1 ∈ A(D).
4.9. Using the results in Exercise 4.7, prove the following elementary properties
of iterated line digraphs: Let D be a digraph. Then
(i) Lk (D) is a digraph with no arcs, for some k, if and only if D is acyclic;
(ii) if D has a pair of cycles joined by a path (possibly of length 0), then
lim nk = ∞,
k →∞
where nk is the order of Lk (D);
(iii) if no pair of cycles of D is joined by a path, then for all sufficiently large
values of k, each connected component of Lk (D) has at most one cycle.
4.10. Prove by induction on k ≥ 1 Proposition 4.5.4.
4.11. Prove Lemma 4.6.1.
4.12. Prove Lemma 4.6.5.
4.13. Prove Lemma 4.6.6.
4.14. Prove Theorem 4.6.7.
4.16 Exercises
225
4.15. Upwards embeddings of MVSP digraphs. Prove that one can embed
every MVSP digraph D into the Cartesian plane such that, if vertices u, v
have coordinates (xu , yu ) and (xv , yv ), respectively, and there is a (u, v)-path
in D, then xu ≤ xv and yu ≤ yv . Hint: consider series composition and
parallel composition separately.
4.16. Prove Proposition 4.7.2. Hint: use induction on the number of reductions
applied for the ‘if’ part and the number of arcs for the ‘only if’ part.
4.17. Prove Proposition 4.7.3.
4.18. Prove part (b) of Lemma 4.8.4. Hint: if u and v are in S then there is a
path from u to v in U G(S). Similarly, if x and y are in S 0 . Use these paths
(corresponding to sequences of non-adjacent vertices in D) to show that if
xu and vy are arcs, then u = v and x = y must hold if D is quasi-transitive.
4.19. (−) Construct an infinite family of path-mergeable digraphs, which are not
in-path-mergeable.
4.20. Prove Proposition 4.10.2.
4.21. (−) Show that the following ‘claim’ is wrong. Let D be a locally insemicomplete digraph and let D contain internally disjoint paths P1 , P2 such
that Pi is an (xi , y)-path (i = 1, 2) and x1 6= x2 . Then x1 and x2 are adjacent.
4.22. Orientations of path-mergeable digraphs. Prove that every orientation
of a path-mergeable digraph is a path-mergeable oriented graph.
4.23. (+) Prove Corollary 4.9.2.
4.24. Path-mergeable digraphs which are neither locally in-semicomplete
nor locally out-semicomplete. Show by a construction that there exists an infinite class of path-mergeable digraphs, none of which is locally
in-semicomplete or locally out-semicomplete. Then extend your construction
to arbitrary degrees of vertex-strong connectivity. Hint: consider extensions.
4.25. (−) Path-mergeable transitive digraphs. Prove that a transitive digraph
D = (V, A) is path-mergeable if and only if for every x, y ∈ V and every pair
xuy, xvy of (x, y)-path of length 2 either u→v or v→u holds.
4.26. Reformulate Lemma 4.10.3 and Theorem 4.10.4 for locally out-semicomplete
digraphs.
4.27. Orientations of locally in-semicomplete digraphs. Prove that every
orientation of a digraph which is locally in-semicomplete is a locally intournament digraph.
4.28. Strong orientations of strong locally in-semicomplete digraphs.
Prove that every strong locally in-semicomplete digraph on at least 3 vertices has a strong orientation.
4.29. Prove Lemma 4.11.2.
4.30. Prove Corollary 4.11.7.
4.31. Prove Theorem 4.11.8.
226
4. Classes of Digraphs
4.32. Recognition of round digraphs. Show that the proof of Theorem 4.11.4
implies a polynomial algorithm to decide whether a digraph D is round and
to find a round labelling of D (if D is round).
4.33. (+) Using Lemma 4.11.13, show that, if D is a non-round decomposable
locally semicomplete digraph, then the independence number of U G(D) is at
most two.
4.34. (−) Give an example of a locally semicomplete digraph on 4 vertices with
no 2-king.
4.35. Prove Proposition 4.11.16.
4.36. Prove the assertion stated in Exercise 4.33 using Lemma 4.11.14 and Proposition 4.11.16.
4.37. Extending in-path-mergeability. Prove that, if P, Q are internally disjoint (x, z)- and (y, z)-paths in an extended locally in-semicomplete digraph
D and no vertex on P − z is similar to a vertex of Q − z, then there is a path
R from either x or y to z in D such that V (R) = V (P ) ∪ V (Q).
4.38. Prove that there exists an O(mn + n2 )-algorithm for checking if a digraph D
with n vertices and m arcs has a decomposition D = R[H1 , . . . , Hr ], r ≥ 2,
where Hi is an arbitrary digraph and the digraph R is either semicomplete
bipartite or connected extended locally semicomplete.
4.39. (−) Let D be a connected digraph which is both quasi-transitive and locally
semicomplete. Prove that D is semicomplete.
4.40. (−) Let D be a connected digraph which is both quasi-transitive and locally
in-semicomplete. Prove that the diameter of U G(D) is at most 2.
4.41. (−) Prove that the intersection number in(D) ≤ n for every digraph D of
order n. Show that this upper bound is sharp (Sen, Das, Roy and West [661]).
4.42. Prove Corollary 4.14.3. Hint: use that each edge is on the boundary of precisely two faces and that each face has at least 3 edges.
4.43. (−) Check which of the following 4 × 4-matrices A = [aij ] have hidden blocktriangular structure (the entries not specified equal zero). Only simultaneous
permutations of rows and columns are allowed.
(a) a1i = i + 1 for i = 1, 2, 3, a2i = a3i = i for i = 2, 3, and a4i = 2 for
i = 2, 3, 4;
(b) a12 = a21 = a14 = a41 = a34 = a43 = 2 and aii = 1 for i = 1, 2, 3, 4.
5. Hamiltonicity and Related Problems
In this chapter we will consider the hamiltonian path and cycle problems for
digraphs as well as some related problems such as the longest path and cycle
problems and the minimum path factor problem. We describe and prove a
number of results in the area as well as formulate several open questions.
We recall that a k-path factor of a digraph D is a collection of k vertexdisjoint paths covering V (D). Recall that the minimum positive integer k
such that D has a k-path factor is the path covering number of D, denoted
by pc(D). A pc(D)-path factor of D is also called a minimum path factor
of D. Recall also that a digraph is traceable if it contains a hamiltonian path.
For arbitrary digraphs the hamiltonian path and hamiltonian cycle problems are very difficult and both are N P-complete (see, e.g. the book [303]
by Garey and Johnson). For convenience of later referencing we state these
results as theorems.
Theorem 5.0.1 The problem to check whether a given digraph has a hamiltonian cycle is N P-complete.
u
t
Theorem 5.0.2 The problem to check whether a given digraph has a hamiltonian path is N P-complete.
u
t
It is worthwhile mentioning that the hamiltonian cycle and path problems
are N P-complete even for some special classes of digraphs. Garey, Johnson
and Tarjan showed [305] that the problem remains N P-complete even for
planar 3-regular digraphs. It follows easily from Theorems 5.0.1 and 5.0.2
that the problem to determine the minimum path factor as well as the longest
path and cycle problems are N P-hard as optimization problems for arbitrary
digraphs. This is also true for several special classes of digraphs. However,
for some important special classes of digraphs these problems are polynomial
time solvable. One such class is the class of acyclic digraphs (see Theorem
2.3.5 and Section 5.3). The reader will see in this chapter that many more
such classes can be found.
In Section 5.1, some powerful necessary conditions, due to Gutin and Yeo,
are considered for a digraph to be hamiltonian. These conditions can be used
for the hamiltonian path problem due to the following simple observation:
228
5. Hamiltonicity and Related Problems
Proposition 5.0.3 A digraph D has a Hamilton path if and only if the digraph D∗ , obtained from D by adding a new vertex x∗ such that x∗ dominates
every vertex of D and is dominated by every vertex of D, is hamiltonian. u
t
In Section 5.2 we prove that the path covering number of an arbitrary
digraph is never more than its independence number. In Section 5.3 we show
that the minimum path factor problem for acyclic digraphs can be solved
quite efficiently. Furthermore, we show that algorithms for finding minimum
path factors in acyclic digraphs are useful in a number of applications.
In Section 5.4, we obtain necessary and sufficient conditions by BangJensen for a path-mergeable digraph to be hamiltonian. Since locally insemicomplete and out-semicomplete digraphs are proper subclasses (see
Proposition 4.10.1) of path-mergeable digraphs, we may use these conditions, in Section 5.5, to derive a characterization of hamiltonian locally insemicomplete and out-semicomplete digraphs. As corollaries, we obtain the
corresponding results for locally semicomplete digraphs. Digraphs with restricted degrees are considered in Section 5.6. There, a number of degreerelated sufficient conditions for a digraph to be hamiltonian are described. In
that section, we also consider a recently introduced and powerful proof technique, called multi-insertion, that can be applied to prove many theorems on
hamiltonian digraphs.
In the last decade quite a number of papers were devoted to studying the
structure of longest cycles and paths of semicomplete multipartite digraphs.
In Section 5.7, we consider the most important results obtained in this area
so far including some striking results by Yeo. The proofs in that section
provide further illustrations of the multi-insertion technique. In Section 5.8,
we discuss generalizations of characterizations of hamiltonian and traceable
extended semicomplete digraphs to extended locally semicomplete digraphs.
Sections 5.9 and 5.10 are devoted to quasi-transitive digraphs. We present
two interesting methods to tackle the hamiltonian path and cycle problems,
and the longest path and cycle problems, respectively, in this class of digraphs. The second method by Bang-Jensen and Gutin allows one to find
even vertex-heaviest paths and cycles in quasi-transitive digraphs in polynomial time (where the weights are on the vertices). The last section is devoted
to results on hamiltonian paths and cycles in some classes of digraphs not considered in the previous sections. The proof of Theorem 5.11.2 by Thomassen
illustrates how the properties of tournaments can be used to prove results on
more general digraphs.
For additional information on hamiltonian and traceable digraphs, see
e.g. the surveys [61, 66] by Bang-Jensen and Gutin, [126] by Bondy, [368] by
Gutin and [728, 729] by Volkmann.
5.1 Necessary Conditions for Hamiltonicity of Digraphs
229
5.1 Necessary Conditions for Hamiltonicity of Digraphs
An obvious condition for a digraph to be hamiltonian is to be strong. Another
obvious and, yet, quite powerful necessary condition for a digraph to be
hamiltonian is the existence of a cycle factor1 . Both conditions can be verified
in polynomial time (see Sections 4.4 and 3.11.4). The purpose of this section is
to describe a series of more powerful conditions, called k-quasi-hamiltonicity,
which were recently introduced by Gutin and Yeo in [379]. An equivalent
form of 1-quasi-hamiltonicity, pseudo-hamiltonicity, was actually investigated
earlier by Babel and Woeginger in [35] for undirected graphs.
We prove that every (k + 1)-quasi-hamiltonian digraph is also k-quasihamiltonian (however, there are digraphs which are k-quasi-hamiltonian, but
not (k + 1)-quasi-hamiltonian). We introduce an algorithm that checks kquasi-hamiltonicity of a given digraph with n vertices and m arcs in time
O(nmk ). Hence, these conditions can be efficiently verified for small values
of k. Thus, they can be incorporated in software systems which investigate
properties of digraphs (or graphs); one such system is described by Delorme,
Ordaz and Quiroz in [189]. We prove that (n − 1)-quasi-hamiltonicity coincides with hamiltonicity and 1-quasi-hamiltonicity is equivalent to pseudohamiltonicity.
5.1.1 Path-Contraction
In this section we consider, for technical reasons, directed multigraphs. We use
a variation of the operation of contraction of a set of vertices in a directed
multigraph. This operation is called path-contraction and is defined as
follows. Let P be an (x, y)-path in a directed multigraph D = (V, A). Then
D//P stands for the directed multigraph with vertex set V (D//P ) = V ∪
{z} − V (P ), where z ∈
/ V , and µD//P (uv) = µD (uv), µD//P (uz) = µD (ux),
µD//P (zv) = µD (yv) for all distinct u, v ∈ V − V (P ). In other words, D//P
is obtained from D by deleting all vertices of P and adding a new vertex
z such that every arc with head x (tail y) and tail (head) in V − V (P )
becomes an arc with head (tail) z and the same tail (head). Observe that
a path-contraction in a digraph results in a digraph (no parallel arcs arise).
We will often consider path-contractions of paths of length one, i.e. arcs e.
Clearly, a directed multigraph D has a k-cycle (k ≥ 3) through an arc e if
and only if D//e has a cycle through z. Observe that the obvious analogue of
path-contraction for undirected multigraphs does not have this nice property
which is of use in this section. The difference between (ordinary) contraction
(which is also called set-contraction) and path-contraction is reflected in
Figure 5.1.
1
Häggkvist [387] posed a problem to find classes of digraphs for which strong
connectivity and the existence of a cycle factor are sufficient for hamiltonicity.
In this chapter we consider some classes with this property.
230
5. Hamiltonicity and Related Problems
a
b
d
x
u
v
y
c
D
a
2
b
a
2
2
d
b
z
z
d
3
c
D/{x, u, v, y}
c
D//P, P = xuvy
Figure 5.1 The two different kinds of contraction, set-contraction and pathcontraction. The integers 2 and 3 indicate the number of corresponding parallel
arcs.
As for set-contraction, for vertex-disjoint paths P1 , P2 , . . . , Pt in D, the
path-contraction D//{P1 , . . . , Pt } is defined as the directed multigraph
(. . . ((D//P1 )//P2 ) . . .)//Pt ; clearly, the result does not depend on the order
of P1 , P2 , . . . , Pt .
5.1.2 Quasi-Hamiltonicity
The results in the remainder of this section are due to Gutin and Yeo. Let
D = (V, A) be a directed multigraph. Let QH1 (D) = (V, A1 ) be the directed
multigraph with arc set
A1 = {e ∈ A : e is contained in a cycle factor of D}.
For k ≥ 2, QHk (D) = (V, Ak ) is the directed multigraph with arc set Ak =
{e ∈ A : QHk−1 (D//e) is strong}. For k ≥ 1, a directed multigraph D is
k-quasi-hamiltonian, if QHk (D) is strong. We assume (by definition) that
every directed multigraph is 0-quasi-hamiltonian. The quasi-hamiltonicity
number of a directed multigraph D of order n, qhn(D), is the maximum
integer k(< n) such that D is k-quasi-hamiltonian.
Figure 5.2 illustrates the notion of quasi-hamiltonicity. The directed multigraph H is 0-quasi-hamiltonian, but not 1-quasi-hamiltonian (QH1 (H) =
H − {(3, 4), (4, 3)} is not strong). Hence, qhn(H) = 0. The directed multigraph D is 1-quasi-hamiltonian as QH1 (D) = D is strong (every arc of D
5.1 Necessary Conditions for Hamiltonicity of Digraphs
231
belongs to a cycle factor of D). However, D is not 2-quasi-hamiltonian since
QH2 (D) is not strong (indeed, QH1 (D//(3, 4)) = QH1 (L) is not strong).
Thus, qhn(D) = 1.
1
2
1
3
2
1
2
3
34
4
6
4
5
H
6
5
6
D
5
L
Figure 5.2 Digraphs.
We start with some basic facts on k-quasi-hamiltonicity.
Proposition 5.1.1 [379] Let D be a directed multigraph of order n(≥ 2) and
let k ∈ {2, 3, . . . , n − 1}. Then A(QHk (D)) ⊆ A(QHk−1 (D)). In particular,
if D is k-quasi-hamiltonian, it is (k − 1)-quasi-hamiltonian.
Proof: We prove the claim by induction on k. Let e ∈ A(QH2 (D)). Thus,
QH1 (D//e) is strong which, in particular, means that D//e has a cycle factor.
Hence, e ∈ A(QH1 (D)). Let now k ≥ 3 and let e ∈ A(QHk (D)). Then,
QHk−1 (D//e) is strong. By the induction hypothesis, QHk−2 (D//e) is also
strong. Hence, e ∈ A(QHk−1 (D)).
u
t
Theorem 5.1.2 [379] A directed multigraph is hamiltonian if and only if it
is (n − 1)-quasi-hamiltonian.
Proof: Clearly every hamiltonian directed multigraph of order 2 is 1-quasihamiltonian. Now assume that all hamiltonian directed multigraphs of order
n − 1 are (n − 2)-quasi-hamiltonian, and let D be a hamiltonian digraph of
order n. Whenever we contract an arc belonging to a hamiltonian cycle we
obtain a hamiltonian digraph of order n − 1, which therefore is (n − 2)-quasihamiltonian. Hence, every arc on a Hamilton cycle lies in QHn−1 (D), which
implies that QHn−1 (D) is strong, i.e. D is (n − 1)-quasi-hamiltonian. Thus,
the ‘only if’ part is proved.
We prove the ‘if’ part. Let D be a directed multigraph, such that
QHn−1 (D) is strong. Let e1 be an arc in QHn−1 (D). Since QHn−2 (D//e1 ) is
strong there exists an arc e2 in QHn−2 (D//e1 ). Since QHn−3 ((D//e1 )//e2 )
232
5. Hamiltonicity and Related Problems
is strong there exists an arc e3 in QHn−3 ((D/e1 )//e2 ). Continuing this procedure we obtain arcs e1 , e2 , . . . , en−2 , such that the directed multigraph
QH1 ((((D//e1 )//e2 ) . . .)//en−2 ) is strong. Let
D0 = (((D//e1 )//e2 ) . . .)//en−2 ,
and observe that, since QH1 (D0 ) is strong and D0 has order 2, D0 must be
hamiltonian. By inserting the arcs e1 , e2 , . . . , en−2 into a Hamilton cycle in
D0 , we obtain a Hamilton cycle in D.
u
t
We leave the proof of the following theorem as a non-trivial exercise (Exercise 5.1).
Theorem 5.1.3 [379] For every k ≥ 0, there exists a digraph D such that
qhn(D) = k < n.
u
t
5.1.3 Pseudo-Hamiltonicity and 1-Quasi-Hamiltonicity
For a positive integer h, a sequence of vertices Q = v1 v2 . . . vhn v1 in a directed
multigraph D of order n is an h-pseudo-hamiltonian walk if every vertex
of D appears h times in the sequence v1 v2 . . . vhn and vi vi+1 ∈ A(D) for
every i = 1, 2, . . . , hn (vhn+1 = v1 ). A directed multigraph D possessing
such a sequence is called h-pseudo-hamiltonian and the minimum h for
which D is h-pseudo-hamiltonian is the pseudo-hamiltonicity number
ph(D) of D. If D has no h-pseudo-hamiltonian walk for any positive integer
h, then ph(D) = ∞. A directed multigraph D is pseudo-hamiltonian if
ph(D) < ∞.
For example, in Figure 5.2, the digraph D is 2-pseudo-hamiltonian:
1212345656431 is a 2-pseudo-hamiltonian walk of D. This digraph is not
1-pseudo-hamiltonian as D is not hamiltonian. Thus, ph(D) = 2. It is not
difficult to see that the digraph H in Figure 5.2 is not pseudo-hamiltonian. We
have already seen that D is 1-quasi-hamiltonian, but H is not. The above
conclusions on pseudo-hamiltonicity of D and H can actually be obtained
from Theorem 5.1.5.
Lemma 5.1.4 follows from the fact that every regular directed multigraph
has a cycle factor (see Exercise 3.70), which implies that every h-regular
directed multigraph can be decomposed into h cycle factors.
Lemma 5.1.4 Every arc of a regular directed multigraph is included in a
cycle factor.
u
t
Theorem 5.1.5 [379] A directed multigraph is pseudo-hamiltonian if and
only if it is 1-quasi-hamiltonian.
5.1 Necessary Conditions for Hamiltonicity of Digraphs
233
Proof: Let D be a pseudo-hamiltonian directed multigraph, let Q be an
h-pseudo-hamiltonian walk in D, and let A(Q) = (v1 v2 , v2 v3 , . . . , vhn−1 vhn ,
vhn v1 ) be the sequence of arcs in Q. Construct a new directed multigraph
H(D, Q) from D by replacing, for every pair x, y with µD (xy) > 0, all arcs
from x to y in D by t(≥ 0) parallel arcs from x to y, where t is the number
of appearances of xy in A(Q). By the definition of an h-pseudo-hamiltonian
walk, H(D, Q) is an h-regular directed multigraph. Thus, by Lemma 5.1.4,
every arc xy in H(D, Q) is in a cycle factor. Therefore, µH(D,Q) (xy) > 0
implies µQH1 (D) (xy) > 0. Since H(D, Q) is strong, we obtain that QH1 (D)
is also strong, i.e. D is 1-quasi-hamiltonian.
Now let D be a 1-quasi-hamiltonian directed multigraph, i.e. QH1 (D) is
strong. For each arc e in QH1 (D) let Fe be a cycle factor in D including e.
Let D0 = ∪e∈A(QH1 (D)) Fe . As the union of cycle factors, D0 is regular. Since
QH1 (D) is strong, D0 is also strong. Therefore, D0 has a eulerian trail, which
corresponds to a pseudo-hamiltonian walk in D.
u
t
The following theorem provides a sharp upper bound for the pseudohamiltonicity number of a digraph.
Theorem 5.1.6 [379] For a pseudo-hamiltonian digraph D, ph(D) ≤ (n −
1)/2. For every integer n ≥ 3, there exists a digraph Hn of order n such that
ph(Hn ) = b(n − 1)/2c.
Proof: Exercise 5.2.
u
t
5.1.4 Algorithms for Pseudo- and Quasi-Hamiltonicity
It is easy to check whether a digraph is 1-quasi-hamiltonian (i.e., by Theorem
5.1.5 is pseudo-hamiltonian). Indeed, checking whether QH1 (D) is strong can
be done in time O(n + m) (see Section 4.4). Hence, it suffices to show how
to verify for each arc xy if this arc is on some cycle factor. We can merely
replace xy by a path xzy, where z is not in D, and check
√ whether the new
digraph has a cycle factor. This can be done√in time O( nm) by Corollary
3.11.7. Thus, we obtain the total time of O( nm2 ). This complexity bound
was improved by Gutin and Yeo [379] as follows.
Theorem 5.1.7 We can check whether a directed multigraph D is pseudohamiltonian in O(nm) time.
Proof: Exercise 5.3.
u
t
The following theorem implies that one can check k-quasi-hamiltonicity
for a constant k in polynomial time.
Theorem 5.1.8 [379] In O(nmk ) time, one can check if a directed multigraph is k-quasi-hamiltonian.
234
5. Hamiltonicity and Related Problems
Proof: In this proof, we describe an algorithm A that, in time T (k), checks
whether a directed multigraph D is k-quasi-hamiltonian. We will show that
T (k) = O(nmk ).
If k = 1, the algorithm A uses the algorithm B of Theorem 5.1.7. Thus,
T (1) = O(nm). If k ≥ 2 then, for each arc e in D, A verifies whether D//e
is (k − 1)-quasi-hamiltonian. The algorithm A forms QHk (D) from all arcs
e such that D//e is (k − 1)-quasi-hamiltonian. Finally, A checks whether
QHk (D) is strong (in time O(m)). This implies that, for k ≥ 2,
T (k) ≤ mT (k − 1) + O(m).
Since T (1) = O(nm), we obtain that T (k) = O(nmk ).
u
t
5.2 Path Covering Number
The following attainable lower bound for the path covering number of a
digraph D is quite trivial: pcc(D) ≤ pc(D). We will see later in this chapter
that pcc(D) = pc(D) for acyclic digraphs and semicomplete multipartite
digraphs D. The aim of this short section is to obtain a less trivial attainable
upper bound for pc(D). This bound is of use in several applications (see, e.g.,
Section 5.3).
Recall that the independence number α(D) of a digraph D is the cardinality of a maximum independent set of vertices of D (a set X ⊆ V (D) is
independent if no pair of vertices in X is adjacent). Rédei’s theorem (Theorem 1.4.5) can be rephrased as saying that every digraph with independence
number 1 has a hamiltonian path and hence path covering number equal 1.
Gallai and Milgram generalized this as follows.
Theorem 5.2.1 (Gallai-Milgram theorem) [298] For every digraph D,
pc(D) ≤ α(D).
This theorem is an immediate consequence of the following lemma by
Bondy [126]:
Lemma 5.2.2 Let D be a digraph and let P = P1 ∪ P2 ∪ . . . ∪ Ps be an s-path
factor of D. Let i(P) (t(P)) denote the set of initial (terminal) vertices of the
paths in P. Suppose that s > α(D). Then there exists an (s − 1)-path factor
P 0 of D such that i(P 0 ) ⊂ i(P) and t(P 0 ) ⊂ t(P).
Proof: The proof is by induction on n, the order of D. The case n = 1
holds vacuously. Let P be as described in the lemma. Let the path Pj in P
be denoted by xj1 xj2 . . . xjrj , j = 1, 2, . . . , s. Since s > α(D) the subdigraph
Dhi(P)i must contain an arc xk1 xj1 for some k 6= j (1 ≤ k, j ≤ s).
If rk = 1, then we can replace Pk , Pj by the path xk1 Pj and obtain the
desired path factor. So suppose that rk > 1. Now consider D∗ = D − xk1 and
5.3 Path Factors of Acyclic Digraphs with Applications
235
the path factor P ∗ which we obtain from P by deleting xk1 from the path Pk .
Clearly α(D∗ ) ≤ α(D) and we have i(P ∗ ) = i(P) − xk1 + xk2 , t(P ∗ ) = t(P).
Thus it follows by the induction hypothesis that D∗ has an (s−1)-path factor
Q such that t(Q) ⊂ t(P ∗ ), i(Q) ⊂ i(P ∗ ).
If xk2 ∈ i(Q), let Qp be the path of Q whose initial vertex is xk2 . Replacing
Qp with xk1 Qp we obtain a path factor in D with the desired properties. So
suppose that xk2 is not an initial vertex of any of the paths in Q. Then xj1
must belong to i(Q) and we obtain the desired path factor by replacing the
path Qr of Q which starts at xj1 by the path xk1 Qr .
u
t
The following theorem due to Erdős and Szekeres [596] follows easily from
Theorem 5.2.1.
Theorem 5.2.3 Let n, p, q be positive integers with n > pq, and let I =
(i1 , i2 , . . . , in ) be a sequence of n distinct integers. Then there exists either
a decreasing subsequence of I with more than p integers or an increasing
subsequence of I with more than q integers.
Proof: Let D = (V, A) be the digraph with V = {i1 , i2 , . . . , in } and A =
{im ik : m < k and im < ik }. Observe the obvious correspondence between
independent sets of D and decreasing subsequences of I (respectively, paths
of D and increasing subsequences of I). Let F = P1 ∪ . . . ∪ Ps be a minimal
path factor of D. By Theorem 5.2.1, s ≤ α(D). Hence, α(D)·maxsj=1 |Pj | ≥
n > pq. Thus, either α(D) > p, i.e., there exists a decreasing subsequence
with α(D) > p integers, or maxsj=1 |Pj | > q, i.e., there exists an increasing
subsequence with more than q integers.
u
t
Very recently, the following improvement on Theorem 5.2.1 in the case of
strong digraphs was proved by Thomassé. This was originally conjectured by
Las Vergnas (see [107]).
Theorem 5.2.4 [695] If a digraph D is strong, then pc(D) ≤ max{α(D) −
1, 1}.
Las Vergnas (see [106]) proved the following generalization of Theorem
5.2.1.
Theorem 5.2.5 Every digraph D of finite out-radius has an out-branching
with at most α(D) vertices of out-degree zero.
u
t
Theorem 5.2.5 implies Theorem 5.2.1 (Exercise 5.7).
5.3 Path Factors of Acyclic Digraphs with Applications
For acyclic digraphs it turns out that the minimum path factor problem can
be solved quite efficiently. This is important since this problem has many
practical applications. One such example is as follows.
236
5. Hamiltonicity and Related Problems
A news agency wishes to cover a set of events E1 , E2 , . . . , En which take
place within the coming week starting at a prescribed time Ti . For each event
Ei its duration time ti and geographical site Oi is known. The news agency
wishes to cover each of these events by having one reporter present for the
full duration of the event. At the same time it wishes to use as few reporters
as possible. Assuming that the travel time tij from Oi to Oj is known for
each 1 ≤ i, j ≤ n, we can model this problem as follows. Form a digraph
D = (V, A) by letting V = {v1 , v2 , . . . , vn } and for every choice of i 6= j put
an arc from vi to vj if Tj ≥ Ti + ti + tij . It is easy to see that D is acyclic.
Furthermore, if the events can be covered by k reporters then D has a k-path
factor (just follow the routes travelled by the reporters). It is also easy to see
that the converse also holds. Hence having an algorithm for the minimum
path factor problem for acyclic digraphs will provide a solution to this and a
large number of similar problems (such as airline and tanker scheduling, see
Exercise 5.8).
Clearly, pc(D) = pcc(D) for every acyclic digraph D. Using flows in
networks, we can effectively find a pcc(D)-path-cycle factor in any digraph D
(see Exercises 3.59 and 3.7). Since a k-path-cycle factor in an acyclic digraph
has no cycles, this implies that the minimum path factor problem for acyclic
digraphs is easy (at least from an algorithmic point of view).
Theorem 5.3.1 For
√ acyclic digraphs the minimum path factor problem is
solvable in time O( nm).
u
t
Another application of the path covering number of acyclic digraphs is
for partial orders. A partial order consists of a set X and a binary relation
‘ ≺ 0 which is transitive (that is, x ≺ y, y ≺ z implies x ≺ z). Let P = (X, ≺)
be a partial order. Two elements x, y ∈ X are comparable if either x ≺ y
or y ≺ x holds. Otherwise x and y are incomparable. A chain in P is
a totally ordered subset Y of X, that is, all elements in Y are pairwise
comparable. An antichain on P is a subset Z of X, no two elements of
which are comparable. Dilworth proved the following famous min-max result
relating chains to antichains:
Theorem 5.3.2 (Dilworth’s theorem) [193] Let P = (X, ≺) be a partial
order. Then the minimum number of chains needed to cover X equals the
maximum number of elements in an antichain.
Proof: Given P = (X, ≺), let D = (X, A) be the digraph such that xy ∈ A
for x 6= y ∈ X if and only if x ≺ y. Clearly, D is transitive. Furthermore,
a path (an independent set) in D corresponds to a chain (antichain) in P .
We need to show that pc(D) = α(D). By Theorem 5.2.1, pc(D) ≤ α(D).
Let F = P1 ∪ P2 ∪ . . . ∪ Pk be a minimum path factor of D. By transitivity
of D, each V (Pi ) induces a complete subgraph in U G(D). Hence, α(D) =
α(U G(D)) ≤ k = pc(D). Thus, pc(D) = α(D).
u
t
5.4 Hamilton Paths and Cycles in Path-Mergeable Digraphs
237
The last theorem can obviously be reformulated as follows: α(D) = pc(D)
for every transitive oriented graph D. We conclude this section with an extension of the analogous result to extended semicomplete digraphs. Lemma
5.3.3 will be used in Section 6.11.
Lemma 5.3.3 Let D be an acyclic extended semicomplete digraph with
α(D) = k, then the following holds:
(a) pc(D) = k.
(b) One can obtain a minimum path factor of D as follows: choose a longest
path P in D, remove V (P ) and continue recursively.
(c) One can find a minimum path factor using the greedy algorithm in (b) in
total time O(n log n) (using the adjacency matrix).
Proof: By Theorem 5.2.1 pc(D) ≤ k. On the other hand no path can contain
two vertices from the same independent set as that would imply that D
contains a cycle. Hence pc(D) = k. To prove (b), let P be a longest path
of D. By the argument above α(D − P ) ≥ k − 1. On the other hand D can
be written as D = S[K a1 , K a2 , . . . , K as ], where S is a semicomplete digraph
and s = |V (S)|. By Rédei’s theorem (Theorem 1.4.5), S has a hamiltonian
path P 0 . In D this path corresponds to a path Q which contains precisely
one vertex from each maximal independent set. Hence Q is a longest path
in D by the remark above and we have α(D − Q) = k − 1. Now the second
claim follows by induction on k. The third claim follows from the description
of procedure MergeHamPathTour in Section 1.9.1, assuming that we have an
adjacency matrix representation of D. Note that we delete the paths as we
find them and hence the total complexity is still O(n log n).
u
t
5.4 Hamilton Paths and Cycles in Path-Mergeable
Digraphs
The class of path-mergeable digraphs was introduced in Section 4.9, where
some of its properties were studied. In this section, we prove a characterization of hamiltonian path-mergeable digraphs due to Bang-Jensen [50].
We begin with a simple lemma which forms the basis for the proof of
Theorem 5.4.2. For a cycle C, a C-bypass is a path of length at least two
with both end-vertices on C and no other vertices on C.
Lemma 5.4.1 [50] Let D be a path-mergeable digraph and let C be a cycle in
D. If D has a C-bypass P , then there exists a cycle in D containing precisely
the vertices V (C) ∪ V (P ).
Proof: Let P be an (x, y)-path. Then the paths P and C[x, y] can be merged
into one (x, y)-path R, which together with C[y, x] forms the desired cycle.
u
t
238
5. Hamiltonicity and Related Problems
Theorem 5.4.2 (Bang-Jensen) [50] A path-mergeable digraph D of order
n ≥ 2 is hamiltonian if and only if D is strong and U G(D) is 2-connected.
Proof: ‘Only if’ is obvious; we prove ‘if’. Suppose that D is strong, U G(D)
is 2-connected and D is not hamiltonian. Let C = u1 u2 . . . up u1 be a longest
cycle in D. Observe that, by Lemma 5.4.1, there is no C-bypass. For each
i ∈ {1, . . . , p} let Xi (respectively Yi ) be the set of vertices of D − V (C) that
can be reached from ui (respectively, from which ui can be reached) by a
path in D − (V (C) − ui ). Since D is strong,
X1 ∪ . . . ∪ Xp = Y1 ∪ . . . ∪ Yp = V (D) − V (C).
Since there is no C-bypass, every path starting at a vertex in Xi and ending
at a vertex in C must end at ui . Thus, Xi ⊆ Yi . Similarly, Yi ⊆ Xi and,
hence, Xi = Yi . Since there is no C-bypass, the sets Xi are disjoint. Since
we assumed that D is not hamiltonian, at least one of these sets, say X1 , is
non-empty. Since U G(D) is 2-connected, there is an arc with one end-vertex
in X1 and the other in V (D) − (X1 ∪ u1 ), and no matter what its orientation
is, this implies that there is a C-bypass, a contradiction.
u
t
Using the proof of this theorem, Lemma 5.4.1 and Proposition 4.9.3, it is
not difficult to show the following (Exercise 5.10):
Corollary 5.4.3 [50] There is an O(nm)-algorithm to decide whether a
given strong path-mergeable digraph has a hamiltonian cycle and find one
if it exists.
Clearly, Theorem 5.4.2 and Corollary 5.4.3 imply an obvious characterization of longest cycles in path-mergeable digraphs and a polynomial algorithm
to find a longest cycle. Neither a characterization nor the complexity of the
hamiltonian path problem for path-mergeable digraphs is currently known.
The following problem was posed by Bang-Jensen and Gutin:
Problem 5.4.4 [65] Characterize traceable path-mergeable digraphs. Is there
a polynomial algorithm to decide whether a path-mergeable digraph is traceable?
For a related result, see Proposition 6.3.2. This result may be considered
as a characterization of traceable path-mergeable digraphs. However, this
characterization seems of not much value from the complexity point of view.
5.5 Hamilton Paths and Cycles in Locally
In-Semicomplete Digraphs
According to Proposition 4.10.1, every locally in-semicomplete digraph is
path-mergeable. By Exercise 5.12, every strong locally in-semicomplete di-
5.5 Hamilton Paths and Cycles in Locally In-Semicomplete Digraphs
239
graph has a 2-connected underlying graph. Thus, Theorem 5.4.2 implies the
following characterization of hamiltonian locally in-semicomplete digraphs2 .
Theorem 5.5.1 (Bang-Jensen, Huang and Prisner) [81] A locally insemicomplete digraph D of order n ≥ 2 is hamiltonian if and only if D is
strong.
u
t
This theorem generalizes Camion’s theorem on strong tournaments (Theorem 1.5.2). Bang-Jensen and Hell [75] showed that for the class of locally
in-semicomplete digraphs Corollary 5.4.3 can be improved to the following
result.
Theorem 5.5.2 [75] There is an O(m + n log n)-algorithm for finding a
hamiltonian cycle in a strong locally in-semicomplete digraph.
In Section 5.4, we remarked that the Hamilton path problem for pathmergeable digraphs is unsolved so far. For a subclass of this class, locally
in-semicomplete digraphs, an elegant characterization, due to Bang-Jensen,
Huang and Prisner, exists.
Theorem 5.5.3 [81] A locally in-semicomplete digraph is traceable if and
only if it contains an in-branching.
Proof: Since a Hamilton path is an in-branching, it suffices to show that
every locally in-semicomplete digraph D with an in-branching T is traceable.
We prove this claim by induction on the number b of vertices of T of in-degree
zero.
For b = 1, the claim is trivial. Let b ≥ 2. Consider a pair of vertices x, y
of in-degree zero in T . By the definition of an in-branching there is a vertex
z in T such that T contains both (x, z)-path P and (y, z)-path Q. Assume
that the only common vertex of P and Q is z.
By Proposition 4.10.2, there is a path R in D that starts at x or y and
terminates at z and V (R) = V (P ) ∪ V (Q). Using this path, we may replace
T with an in-branching with b − 1 vertices of in-degree zero and apply the
induction hypothesis of the claim.
u
t
Clearly, Theorem 5.5.3 implies that a locally out-semicomplete digraph is
traceable if and only if it contains an out-branching. By Proposition 1.6.1,
we have the following:
Corollary 5.5.4 A locally in-semicomplete digraph is traceable if and only
if it contains only one terminal strong component.
u
t
2
Actually, this characterization, as well as the other results of this section, were
originally proved only for oriented graphs. However, as can be seen from Exercises
4.27 and 4.28, the results for oriented graphs immediately imply the results of
this section.
240
5. Hamiltonicity and Related Problems
Using Corollary 5.5.4, Bang-Jensen and Hell [75] proved the following:
Theorem 5.5.5 A longest path in a locally in-semicomplete digraph D can
be found in time O(m + n log n).
u
t
Corollary 5.5.4 and Lemma 4.10.3 imply the following:
Corollary 5.5.6 (Bang-Jensen) [44] A locally semicomplete digraph has
a hamiltonian path if and only if it is connected.
u
t
Notice that there is a nice direct proof of this corollary (using Proposition
4.10.2), which is analogous to the classical proof of Rédei’s theorem displayed
in procedure HamPathTour in Section 1.9.1. See Exercise 5.14.
5.6 Hamilton Cycles and Paths in Degree-Constrained
Digraphs
In Subsection 5.6.1 we formulate certain sufficient degree-constrained conditions for hamiltonicity of digraphs. Several of these conditions do not follow
from the others, i.e. there are certain digraphs that can be proved to be
hamiltonian using some condition but none of the others. (The reader will be
asked to show this in the exercises.)
In Subsection 5.6.3 we provide proofs to some of these conditions to illustrate the power of a recently introduced approach, which we call the multiinsertion technique. (This technique can be traced back to Ainouche [9] for
undirected graphs and to Bang-Jensen [48] for digraphs, see also the paper
[68] by Bang-Jensen, Gutin and Huang). The technique itself is introduced
in Subsection 5.6.2. The strength of the multi-insertion technique lies in the
fact that we can prove the existence of a hamiltonian cycle without actually
exhibiting it. Moreover, our hamiltonian cycles may have quite a complicated
structure. For example, compare the hamiltonian cycles in the proof of Theorem 5.6.1 to the hamiltonian paths constructed in the inductive proof of
Theorem 1.4.5. The multi-insertion technique is used in some other parts of
this book, see e.g. Section 5.7.
Let x, y be a pair of distinct vertices in a digraph D. The pair {x, y} is
dominated by a vertex z if z→x and z→y; in this case we say that the
pair {x, y} is dominated. Likewise, {x, y} dominates a vertex z if x→z
and y→z; we call the pair {x, y} dominating.
5.6.1 Sufficient Conditions
Considering the converse digraph and using Theorem 5.5.1, we see that a
locally out-semicomplete digraph is hamiltonian if and only if it is strong.
This can be generalized as follows. We prove Theorem 5.6.1 in Subsection
5.6.3.
5.6 Hamilton Cycles and Paths in Degree-Constrained Digraphs
241
Theorem 5.6.1 (Bang-Jensen, Gutin and Li) [69] Let D be a strong digraph of order n ≥ 2. Suppose that, for every dominated pair of non-adjacent
vertices {x, y}, either d(x) ≥ n and d(y) ≥ n−1 or d(x) ≥ n−1 and d(y) ≥ n.
Then D is hamiltonian.
The following example shows the sharpness of the conditions of Theorem
5.6.1 (and Theorem 5.6.5), see Figure 5.3. Let G and H be two disjoint
transitive tournaments such that |V (G)| ≥ 2, |V (H)| ≥ 2. Let w be the vertex
of out-degree 0 in G and w0 the vertex of in-degree 0 in H. Form a new digraph
by identifying w and w0 to one vertex z. Add four new vertices x, y, u, v
and the arcs {xv, yv, ux, uy} ∪ {xz, zx, yz, zy} ∪ {rg : r ∈ {x, y, v}, g ∈
V (G) − w} ∪ {hs : h ∈ V (H) − w0 , s ∈ {u, x, y}}. Denote the resulting
digraph by Qn , where n is the order of Qn . It is easy to check that Qn
is strong and non-hamiltonian (Exercise 5.17). Also x, y is the only pair of
non-adjacent vertices which is dominating (dominated, respectively). An easy
computation shows that
d(x) = d(y) = n − 1 = d+ (x) + d− (y) = d− (x) + d+ (y).
v
y
x
u
z
G−w
H − w0
Figure 5.3 The digraph Qn . The two unoriented edges denote 2-cycles.
Combining Theorem 5.6.1 with Proposition 5.0.3 one can obtain sufficient
conditions for a digraph to be traceable (see also Exercise 5.16). Theorem
5.6.1 also has the following immediate corollaries.
Corollary 5.6.2 (Ghouila-Houri) [315] If the degree of every vertex in a
strong digraph D of order n is at least n, then D is hamiltonian.
u
t
Corollary 5.6.3 Let D be a digraph of order n. If the minimum semi-degree
of D, δ 0 (D) ≥ n/2, then D is hamiltonian.
u
t
It turns out that even a slight relaxation of Corollary 5.6.3 brings in nonhamiltonian digraphs. In particular, Darbinyan [177] proved the following:
242
5. Hamiltonicity and Related Problems
Proposition 5.6.4 Let D be a digraph of even order n ≥ 4 such that the
degree of every vertex of D is at least n − 1 and δ 0 (D) ≥ n/2 − 1. Then
either D is hamiltonian or D belongs to a non-empty finite family of nonhamiltonian digraphs.
u
t
By Theorem 5.5.1, a locally semicomplete digraph is hamiltonian if and
only if it is strong [44]. This result was generalized by Bang-Jensen, Gutin
and Li [69] as follows.
Theorem 5.6.5 Let D be a strong digraph of order n. Suppose that D satisfies min{d+ (x) + d− (y), d− (x) + d+ (y)} ≥ n for every pair of dominating
non-adjacent and every pair of dominated non-adjacent vertices {x, y}. Then
D is hamiltonian.
We prove this theorem in Subsection 5.6.3. Theorem 5.6.5 implies Corollary 5.6.3 as well as the following theorem by Woodall [739]:
Corollary 5.6.6 Let D be a digraph of order n ≥ 2. If d+ (x) + d− (y) ≥ n
for all pairs of vertices x and y such that there is no arc from x to y, then
D is hamiltonian.
u
t
The following theorem generalizes Corollaries 5.6.2, 5.6.3 and 5.6.6. The
↔
inequality of Theorem 5.6.7 is best possible: Consider K n−2 (n ≥ 5) and fix
↔
a vertex u in this digraph. Construct the digraph Hn by adding to K n−2 a
↔
pair v, w of vertices such that both v and w dominate every vertex in K n−2
and are dominated by only u, see Figure 5.4. It is easy to see that Hn is
strong and non-hamiltonian (Hn − u is not traceable). However, v, w is the
only pair of non-adjacent vertices in Hn and d(v) + d(w) = 2n − 2.
v
w
u
↔
K n−3
Figure 5.4 The digraph Hn .
Theorem 5.6.7 (Meyniel’s theorem) [564] Let D be a strong digraph of
order n ≥ 2. If d(x) + d(y) ≥ 2n − 1 for all pairs of non-adjacent vertices in
D, then D is hamiltonian.
u
t