Search

A graph is clique-perfect if the maximum size of a clique-independent set (a set of pairwise disjoint maximal cliques) and the minimum size of a clique-transversal set (a set of vertices meeting every maximal clique) coincide for each induced subgraph. A graph is balanced if its clique-matrix contains no square submatrix of odd size with exactly two ones per row and column. In this work, we give linear-time recognition algorithms and minimal forbidden induced subgraph characterizations of clique-perfectness and balancedness ofP4-tidy graphs and a linear-time algorithm for computing a maximum clique-independent set and a minimum clique-transversal set for anyP4-tidy graph. We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs. Finally, we show that clique-perfectness of diamond-free graphs can be decided in polynomial time by showing that a diamond-free graph is clique-perfect if and only if it is balanced. [ABSTRACT FROM PUBLISHER]

Abstract: The clique-transversal number τ c(G) of a graph G is the minimum size of a set of vertices meeting all the cliques. The clique-independence number α c(G) of G is the maximum size of a collection of vertex-disjoint cliques. A graph is clique-perfect if these two numbers are equal for every induced subgraph of G. Unlike perfect graphs, the class of clique-perfect graphs is not closed under graph complementation nor is a characterization by forbidden induced subgraphs known. Nevertheless, partial results in this direction have been obtained. For instance, in [Bonomo, F., M. Chudnovsky and G. Durán, Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs, Discrete Appl. Math. 156 (2008), pp. 1058–1082], a characterization of those line graphs that are clique-perfect is given in terms of minimal forbidden induced subgraphs. Our main result is a characterization of those complements of line graphs that are clique-perfect, also by means of minimal forbidden induced subgraphs. This implies an O(n2) time algorithm for deciding the clique-perfectness of complements of line graphs and, for those that are clique-perfect, finding α c and τ c. [Copyright &y& Elsevier]

Abstract: A graph has the Kőnig property if its matching number equals its transversal number. Lovász proved a characterization of graphs having the Kőnig property by forbidden subgraphs, restricted to graphs with a perfect matching. Korach, Nguyen, and Peis proposed an extension of Lovászʼs result to a characterization of all graphs having the Kőnig property in terms of forbidden configurations (certain arrangements of a subgraph and a maximum matching). In this work, we prove a characterization of graphs having the Kőnig property in terms of forbidden subgraphs which is a strengthened version of the characterization by Korach et al. As a consequence of our characterization of graphs with the Kőnig property, we prove a forbidden subgraph characterization for the class of edge-perfect graphs. [Copyright &y& Elsevier]

Abstract: Probe (unit) interval graphs form a superclass of (unit) interval graphs. A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe interval graphs were introduced by Zhang for an application concerning with the physical mapping of DNA in the human genome project. In this work, we present characterizations by minimal forbidden induced subgraphs of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. [Copyright &y& Elsevier]

Abstract: A graph is balanced if its clique-vertex incidence matrix is balanced, i.e., it does not contain a square submatrix of odd order with exactly two ones per row and per column. Interval graphs, obtained as intersection graphs of intervals of a line, are well-known examples of balanced graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. Circular-arc graphs generalize interval graphs, but are not balanced in general. In this work we characterize balanced graphs by minimal forbidden induced subgraphs restricted to graphs that belong to some classes of circular-arc graphs. [ABSTRACT FROM AUTHOR]

Abstract: A {0, 1}-matrix is balanced if it contains no square submatrix of odd order with exactly two 1''s per row and per column. Balanced matrices lead to ideal formulations for both set packing and set covering problems. Balanced graphs are those graphs whose clique-vertex incidence matrix is balanced. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to some graph classes which also lead to polynomial time or even linear time recognition algorithms within the corresponding subclasses. [Copyright &y& Elsevier]

Abstract: Circular-arc graphs are the intersection graphs of open arcs on a circle. Circle graphs are the intersection graphs of chords on a circle. These graph classes have been the subject of much study for many years and numerous interesting results have been reported. Many subclasses of both circular-arc graphs and circle graphs have been defined and different characterizations formulated. In this survey, we summarize the most important structural results related to circular-arc graphs and circle graphs and present the main open problems. [Copyright &y& Elsevier]

Abstract: A graph is balanced if its clique-matrix contains no edge–vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs that belong to one of the following graph classes: complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs. These characterizations lead to linear-time recognition algorithms for balanced graphs within the same three graph classes. [Copyright &y& Elsevier]

We initiate the study of graph classes of power-bounded clique-width, that is, graph classes for which there exist integers k and ℓ such that the k th powers of the graphs are of clique-width at most ℓ . We give sufficient and necessary conditions for this property. As our main results, we characterize graph classes of power-bounded clique-width within classes defined by either one forbidden induced subgraph, or by two connected forbidden induced subgraphs. We also show that for every positive integer k , there exists a graph class such that the k th powers of graphs in the class form a class of bounded clique-width, while this is not the case for any smaller power. [ABSTRACT FROM AUTHOR]