Perfect Digraph Research Articles

Strong and weak Perfect Digraph Theorems for perfect, \alpha-perfect and strictly perfect digraphs

Strong and weak Perfect Digraph Theorems for perfect, <i>\alpha</i>-perfect and strictly perfect digraphs

Richardson’s theorem in quasi-transitive and pre-transitive digraphs

A subset N of V(D) is said to be a kernel if it satisfies the following two properties: (1) for any two different vertices x and y in N there is no arc between them, and (2) for each vertex u in V(D)$$\setminus N$$ there exists v in N such that (u,v) $$\in$$ A(D). If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. In Galeana-Sanchez and Rojas-Monroy (Discrete Math, 275: 129–136, 2004) and Galeana-Sanchez and Rojas-Monroy (Discrete Math. 306: 1969–1974, 2006) the authors establish sufficient conditions to guarantee the kernel perfectness in digraphs, possibly infinite, where their set of arcs can be partitioned into at most two pre-transitive (resp. quasi-transitive) digraphs. In the present paper we consider those, also possibly infinite, digraphs where the set of arcs can be partitioned into at least three quasi-transitive (resp. pre-transitive) digraphs, and establish sufficient conditions to guarantee the kernel perfectness. In both cases we derive Richardson’s theorem, which states that every finite digraph without cycles of odd length has a kernel.

A semi-strong perfect digraph theorem

Reed (1987) showed that, if two graphs are P4-isomorphic, then either both are perfect or none of them is. In this note, we will derive an analogous result for perfect digraphs.

Unions of digraphs which become kernel perfect

A kernel of a digraph, D, is a set of vertices, K, which is both independent: every two vertices are not adjacent; and absorbent: for every vertex in V(D)∖K there exists an ex-neighbor in K. A digraph is said to be kernel-perfect if and only if any induced subdigraph has a kernel. A classical result obtained by Sands, Sauer and Woodrow in 1982 asserts that the union of two transitive digraphs is a kernel-perfect digraph. A digraph D is transitive whenever the existence of the arcs (u,v) and (v,w) in D implies the existence of the arc (u,w) in D. Later, it was studied the union of a right-pretransitive digraph and a left-pretransitive digraph, and how it becomes a kernel-perfect digraph (Galeana-Sánchez and Rojas-Monroy, 2004). In this paper we obtain sufficient conditions for the union of two kernel-perfect digraphs becomes a kernel perfect digraph. And these conditions are tight. As a consequence many previous results are generalized.

Infinite kernel perfect digraphs

Let D be a digraph, possibly infinite, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A subset K of V(D) is said to be a kernel if it is both independent (a vertex in K has no successor in K) and absorbing (a vertex not in K has a successor in K). An infinite digraph D is said to be a finitely critical kernel imperfect digraph if D contains no kernel but every finite induced subdigraph of D contains a kernel. In this paper we will characterize the infinite kernel perfect digraphs by means of finitely critical imperfect digraphs and strong components of its asymmetric part and then, by using some previous theorems for infinite digraphs, we will deduce several results from the main result. Richardson’s theorem establishes that if D is a finite digraph without cycles of odd length, then D has a kernel. In this paper we will show a generalization of this theorem for infinite digraphs.

CKI-Digraphs, Generalized Sums and Partitions of Digraphs

A kernel of a digraph is a set of vertices which is both independent and absorbent. Let $$D$$D be a digraph such that every proper induced subdigraph of $$D$$D has a kernel; $$D$$D is said to be kernel perfect digraph (KP-digraph) if the digraph $$D$$D has a kernel and critical kernel imperfect digraph (CKI-digraph) if the digraph $$D$$D does not have a kernel. We characterize the CKI-digraphs with a partition into an independent set and a semicomplete digraph. The generalized sum $$G(F_u)$$G(Fu) of a family of mutually disjoint digraphs $$\{F_u\}_{u\in V(G)}$${Fu}u?V(G) over a graph $$G$$G is a digraph defined as follows: Consider $$\cup _{u\in V(G)}F_u$$?u?V(G)Fu, and for each $$x\in V(F_v)$$x?V(Fv) and $$y\in V(F_w)$$y?V(Fw) with $$\{v,w\}\in E(G)$${v,w}?E(G) choose exactly one of the two arcs $$(x,y)$$(x,y) or $$(y,x)$$(y,x). We characterize the asymmetric CKI-digraphs which are generalized sums over an edge or a cycle. Furthermore, we give sufficient conditions on $$G$$G and the family $$\{F_u\}_{u\in V(G)}$${Fu}u?V(G), such that the generalized sum $$G(F_u)$$G(Fu) has a kernel or is a is KP-digraph.

Perfect Digraphs

AbstractThe clique number of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number . Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is ‐complete to decide whether a perfect digraph has a kernel.

Characterization of asymmetric CKI- and KP-digraphs with covering number at most 3

A set N⊆V(D) is said to be a kernel if N is an independent set and for every vertex x∈(V(D)∖N) there is a vertex y∈N such that xy∈A(D). Let D be a digraph such that every proper induced subdigraph of D has a kernel. D is said to be kernel perfect digraph (KP-digraph) if the digraph D has a kernel and critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. In this paper we characterize the asymmetric CKI-digraphs with covering number at most 3. Moreover, we prove that the only asymmetric CKI-digraphs with covering number at most 3 are: C⃗3, C⃗5 and C⃗7(1,2). Several interesting consequences are obtained.

Some sufficient conditions for the existence of kernels in infinite digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V ( D ) − N there exists an arc from w to N . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D , a semikernel modulo F of D is an independent set of vertices S of D such that for every z ∈ V ( D ) − S for which there exists an ( S , z ) -arc of D − F , there also exists an ( z , S ) -arc in D . In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F . As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.

A note on Duchet's conjecture

A kernel in a digraph is a set of vertices that is both an absorbing and independent set. In [4] P. Duchet conjectured that a digraph such that every odd circuit has two short chords is kernel perfect digraph (i.e. every induced subdigraph has a kernel) and he proved the conjecture for digraphs where every 3 -circuit is reversal. We give another simple proof of this result by using reorientation and constructive methods. We also show that the conjecture is valid for digraphs D where the underlying undirected spanning subgraph deduced from the spanning subdigraph of D with only pairs of symmetric arcs is a comparability graph.

Kernel perfect and critical kernel imperfect digraphs structure

Abstract A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V ( D ) − N there exists an arc from w to N . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If F is a set of arcs of D , a semikernel modulo F , S of D is an independent set of vertices of D such that for every z ∈ V ( D ) − S for which there exists an Sz− arc of D − F , there also exists an zS− arc in D . In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented.

New sufficient conditions for the existence of kernels in digraphs

Abstract A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V ( D ) − N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If F is a set of arcs of D, a semikernel modulo F, S of D is an independent set of vertices of D such that for every z ∈ V ( D ) − S for which there exists an S z − arc of D − F , there also exists an z S − arc in D. In this work new sufficient conditions for a digraph to be a critical kernel imperfect digraph, in terms of semikernel modulo F, are presented.

Kernel perfect and critical kernel imperfect digraphs structure

A kernel $N$ of a digraph $D$ is an independent set of vertices of $D$ such that for every $w \in V(D)-N$ there exists an arc from $w$ to $N$. If every induced subdigraph of $D$ has a kernel, $D$ is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If $F$ is a set of arcs of $D$, a semikernel modulo $F$, $S$ of $D$ is an independent set of vertices of $D$ such that for every $z \in V(D)- S$ for which there exists an $Sz-$arc of $D-F$, there also exists an $zS-$arc in $D$. In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented.

A Note on Perfect Elimination Digraphs

The purpose of this note is to settle a conjecture on perfect elimination digraphs raised by Haskins and Rose in this journal, and to make some remarks that illuminate the nature of digraphs satisfying the conditions defined by them.