Abstract
Graph Theory A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.
Highlights
Two fundamental combinatorial optimization problems are set packing and set covering, which can be expressed by max cT x s.t
The matrix A is balanced if all its submatrices are both perfect and ideal or, equivalently, if it contains no submatrix of odd order with exactly two ones per row and per column [2, 18]
The minimal forbidden induced subgraphs of perfect graphs are the chordless cycles of odd length having at least 5 vertices, called odd holes C2k+1, and their complements, the odd antiholes C2k+1
Summary
Two fundamental combinatorial optimization problems are set packing and set covering, which can be expressed by max cT x s.t. The minimal forbidden induced subgraphs of perfect graphs are the chordless cycles of odd length having at least 5 vertices, called odd holes C2k+1, and their complements, the odd antiholes C2k+1. No balanced graph contains an odd hole, odd antihole, or any pyramid as induced subgraph. Theorem 1 ([2, 9]) A graph is balanced if and only if it has no unbalanced cycle, or, equivalently, if and only if it contains no induced extended odd sun. Partial answers are obtained in [10] where minimal forbidden induced subgraph characterizations of balanced graphs restricted to the following graph classes are found: P4-tidy graphs, paw-free graphs, line graphs, and complements of line graphs.
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