A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge’s Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd, eb}. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect...

A graph G is clique-perfect if the cardinality of a maximum cliqueindependent set of H is equal to the cardinality of a minimum cliquetransversal of H, for every induced subgraph H of G. When equality holds for every clique subgraph of G, the graph is c–clique-perfect. A graph G is K-perfect when its clique graph K(G) is perfect. In this work, relations are described among the classes of perfec...

Boros and Gurvich [3] showed that every clique-acyclic superorientation of a perfect graph has a kernel. We prove the following extension of their result: if G is an h-perfect graph, then every clique-acyclic and odd-hole-acyclic superorientation of G has a kernel. We propose a conjecture related to Scarf’s Lemma that would imply the reverse direction of the Boros-Gurvich theorem without relyin...

The semirandom graph process is a single player game in which the initially presented an empty on $n$ vertices. In each round, vertex $u$ to independently and uniformly at random. then adaptively selects $v$ adds edge $uv$ graph. For fixed monotone property, objective of force satisfy this property with high probability as few rounds possible. We focus problem constructing perfect matching part...

A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subprob-lems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, ...

Weakly s-arc transitive graphs are introduced and determined. A graph is said to be weakly s-arc transitive if its endomorphism monoid acts transitively on the set of s-arcs. The main results are: (1) A nonbipartite graph is weakly s-arc transitive if and only if it is s-arc transitive. (2) A tree with diameter d is weakly s-arc transitive for all 0 s d. (3) A bipartite graph with girth g = 2s ...

For a graph G=(V,E) with V=V(G) and E=E(G), perfect double Italian dominating function is f:V→{0,1,2,3} having the property that 3≤∑u∈NG[v]f(u)≤4, for every vertex v∈G f(v)∈{0,1}. The weight of f sum f(V)=∑v∈V(G)f(v) minimum on G domination number γdIp(G) G. We initiate study functions. check γdIp some standard graphs evaluate γdI such graphs. functions versus Roman are perused. NP-completeness...