On a class of kernel-perfect and kernel-perfect-critical graphs

Chilakamarri, K.B. and P. Hamburger, On a class of kernel-perfect and kernel-perfect-critical graphs, Discrete Mathematics 118 (1993) 253-257. In this note we present a construction of a class of graphs in which each of the graphs is either kernel-perfect or kernel-perfect-critical. These graphs originate from the theory of games (Von Neumann and Morgenstern). We also find criteria to distinguish kernel-perfect graphs from kernelperfect-critical graphs in this class. We obtain some of the previously known classes of kernelperfect-critical graphs as special cases of the present construction given here. The construction that we give enlarges the class of kernel-perfect-critical graphs. Let G=(X, W) be a directed graph with no loops and no multiple arcs. For any subsetKcX,onedefinesr-(K)={xEX:thereisanarcfromxtoyforsomeyinK}. If a vertex x is in r-(K), then we say that x is absorbed by K. Throughout this note we consider graphs with no loops and no multiple arcs. A kernel in a directed graph G =(X, W) is a subset Kc X of vertices with the following properties: Correspondence to: Peter Hamburger, Indiana University-Purdue University at Fort Wayne, 2101 Coliseum Boulevard East, Fort Wayne, IN 46805-1499, USA. 0012-365X/93/$06.00 8 1993-Elsevier Science Publishers B.V. All rights reserved (1) K is stable (or independent): r-( K)n K =@, and (2) K is dominant (or absorbant): r (K) u K = X. A graph G=( X, W) is called kernel-perfect gruph (KP) if every induced subgraph (including G) has a kernel. A graph G is called kernel-perjbct-criticul graph (KPC) if G has no kernel, but every proper induced subgraph of G has a kernel. These graphs have been studied by Galeana-Sanchez and Neumann-Lara [3] and Duchet and Meyniel [4]. For a more complete list of references see [3]. We write (x, y) for an asymmetric arc from x to y and [x, y] when both (x, y) and (y, x) are present in the graph. We call [x, y] a symmetric arc and indicate the symmetric arcs in the figures by arcs with no arrows. We present a construction which significantly enlarges the class of KP and KPC graphs. We also present a condition which distinguishes the KP graphs from the KPC graphs for the graphs obtained from the construction. We prove several corollaries showing that some of the previously known classes of KPC graphs (GaleanaSanchez and Neumann-Lara [3]) are special cases of the present construction. The following result was proved by Galeana-Sanchez and Neumann-Lara [3], Duchet and Meyniel [4] and by Duchet [S]. Lemma 1. JfAsym( G) is acyclic, then G is u KP graph, where Asym( G) is the sutxgraph spanned by asymmetric arcs in G. Now we proceed to construct a class of graphs in which each of the graphs is either a KP or a KPC graph. In the theorem that follows the construction, we obtain a condition that distinguishes the KP graphs from the KPC graphs. Construction. Let n, r and s be integers with r > 1, s > 0 and 2r + 2s + 1 <II. Let c, be a directed cycle of length n on vertices (x0,x1, . ,_x,_r ), the directed arcs being (%,.~I), (X,,XZ), (%,Xg), . . ..(-q2,x,_1),(x,_1,x0). Let the graph G(n,r,s) be constructed from 2, by adding symmetric arcs [-Ui, xj] whenever the indices i and j satisfy the following condition: k(i-,j)-r+ l,r+2, . . . . r+s (modn). If s=O, then no symmetric arcs are added and resulting graph is simply a directed cycle. We assume s3 1 in what follows. This completes the construction. In Figs. 1 and 2 we have two examples. The Fig. 1 is the graph G(7,2,1) and Fig. 2 is thegraphG(7,1,1).Thesetjx,,x,)’ IS a kernel in the Fig. 1. There is no kernel in the graph of Fig. 2. Remark 2. If the graph G(n, r, s) has a kernel, then it is a KP graph; otherwise it is a KPC graph.