Monday , September 26 2022

Some Combinatorial Optimization Problems for Weak-Bisplit Graphs

University of Bacău, Department of Mathematics and Informatics
8 Spiru Haret, Bacău, Romania

University of Bacău, Department of Mathematics and Informatics
8 Spiru Haret, Bacău, Romania

Abstract: There exists linear algorithms to recognize weak-bisplit graphs and NP-complete optimization problems are efficiently resolved for this class of graphs. In this paper, using weak-decomposition, we give necessary a sufficient conditions for a graph to be weak-bisplit, bi-cograph, weak-bisplit cograph. We also give an algorithm O(n+m) to determine, the density of a weak-bisplit cograph and we calculate directly the domination number for this class of graphs.

Keywords: Weak-bisplit graphs, Bi-cographs, Weak-bisplit cographs.

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Mihai TĂLMACIU, Elena NECHITA, Some Combinatorial Optimization Problems for Weak-Bisplit Graphs, Studies in Informatics and Control, ISSN 1220-1766, vol. 19 (2), pp. 427-434, 2010.

1. Introduction

Throughout this paper G=(V,E) is a simple (i.e. finite, undirected, without loops and multiple edges) graph [2]. Let coImage1117 denote the complement graph of G. For UImage1118V let G(U) denote the subgraph of G induced by U. By GX we mean the graph G(VX), whenever XImage1118V, but we often denote it simply by Gv (Image1119V) when there is no ambiguity. If vImage1069V is a vertex in G, the neighborhood NG(n)denotes the vertices of Gv that are adjacent to v. We write N(v) when the graph G appears clearly from the context. The neighborhood of the vertex v in the complement of the graph G is denoted by Image1120. For any subset S of vertices in the graph G the neighborhood of S is Image1121 and N[S]=SImage1122N(S). A clique is a subset of V with the property that all the vertices are pairwise adjacent. The clique number (density) of G, denoted by Image1123(G) is the cardinal of the maximum clique. A clique cover is a partition of the vertices set such that each part is a clique. Image1124(G) is the cardinal of a smallest possible clique cover of G; it is called the clique cover number of G. A stable (or independent) set is a subset of V with the property that all the vertices are pairwise non-adjacent. The stability number of G is Image1125(G)=Image1126; the chromatic number of G is Image1127(G)=Image1128.

A dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is joined to at least one member of D by some edge. The domination number γ(G) is the number of vertices in a smallest dominating set for G.

By we mean a chordless path on nImage11293 vertices, the chordless cycle on nImage11303 vertices, and the complete graph on nImage11301 vertices. If e=xyImage1131E, we also denote x~y; we also denote xy whenever x, y are not adjacent in G. A set A is totally adjacent (non adjacent) with a set B of vertices (AImage1132B=Image1133) if ab is (is not) edge, for any a vertex in A and any b vertex in B; we note denote A~B (A≁B). A graph G is F-free if none of its induced subgraphs is in F.

The subset AImage1134V is called a cutset if GA is not connected. If, in addition, none of the proper subsets of A is a cutest, then A is called a minimal cutset .

The paper is organized as follows. In Section 2 we give preliminary results. In Section 3 we give a characterization of weak-bisplit graphs.


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