Based on a new data structure called PATCH Graph, an eecient completion procedure for nitely presented groups is described. A PATCH Graph represents rules and their symmetrized forms as cycles in a Cayley graph structure. Completion is easily performed directly on the graph, and structure sharing is enforced. The structure of the graph allows us to avoid certain redundant inferences. The PATCH ...

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...

In this paper, we verify the diameter of zero divisor graphs with respect to direct product. Keywords—Atomic lattice, complement of graph, diameter, direct product of lattices, 0-distributive lattice, girth, product of graphs, prime ideal, zero divisor graph.

A graph is perfect if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called basic. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak...

Obstacle representations of graphs have been investigated quite intensely over the last few years. We focus on graphs that can be represented by a single obstacle. Given a (topologically open) polygon C and a finite set P of points in general position in the complement of C, the visibility graph GC(P ) has a vertex for each point in P and an edge pq for any two points p and q in P that can see ...

Generalizing the star graphs of M\"uller-Stach and Westrich, we describe a class whose associated graph hypersurface is equipped with non-trivial torus action. For such graphs, show that Euler characteristic corresponding projective complement zero. In contrast, also in question can take any integer value for suitable graph. This disproves conjecture Aluffi strong sense.

We assume that the reader is familiar with standard graph-theoretic terminology and refer the readers to Bondy and Murty (2008) for any concept and notation that is not defined here. In this paper, we consider simple, undirected graphs. Given two graphsG andH , the Ramsey numberR(G,H) is the smallest integer n such that every graph F on n vertices contains a copy of G, or its complement F conta...