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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: 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]

The matching number of a graph is the maximum size of a set of vertex-disjoint edges. The transversal number is the minimum number of vertices needed to meet every edge. A graph has the König–Egerváry property if its matching number equals its transversal number. Lovász proved a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs within graphs with a perfect matching. Korach, Nguyen, and Peis proposed an extension of Lovász’s result to a characterization of all graphs having the König–Egerváry property in terms of forbidden configurations (which are certain arrangements of a subgraph and a maximum matching). In this work, we prove a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs which is a strengthened version of the characterization by Korach et al. Using our characterization of graphs with the König–Egerváry property, we also prove a forbidden subgraph characterization for the class of edge-perfect graphs. [ABSTRACT FROM AUTHOR]

Abstract: A circle graph is the intersection graph of a family of chords on a circle. There is no known characterization of circle graphs by forbidden induced subgraphs that do not involve the notions of local equivalence or pivoting operations. We characterize circle graphs by a list of minimal forbidden induced subgraphs when the graph belongs to one of the following classes: linear domino graphs, -tidy graphs, and tree-cographs. We also completely characterize by minimal forbidden induced subgraphs the class of unit Helly circle graphs, which are those circle graphs having a model whose chords have all the same length, are pairwise different, and satisfy the Helly property. [Copyright &y& Elsevier]