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A family of sets is (p , q) - intersecting if every nonempty subfamily of p or fewer sets has at least q elements in its total intersection. A family of sets has the (p , q) - Helly property if every nonempty (p , q) -intersecting subfamily has total intersection of cardinality at least q. The (2 , 1) -Helly property is the usual Helly property. A hypergraph is (p , q) - Helly if its edge family has the (p , q) -Helly property and hereditary (p , q) -Helly if each of its subhypergraphs has the (p , q) -Helly property. A graph is (p , q) - clique-Helly if the family of its maximal cliques has the (p , q) -Helly property and hereditary (p , q) -clique-Helly if each of its induced subgraphs is (p , q) -clique-Helly. The classes of (p , q) - biclique-Helly and hereditary (p , q) -biclique-Helly graphs are defined analogously. In this work, we prove several characterizations of hereditary (p , q) -Helly hypergraphs, including one by minimal forbidden partial subhypergraphs. On the algorithmic side, we give an improved time bound for the recognition of (p , q) -Helly hypergraphs for each fixed q and show that the recognition of hereditary (p , q) -Helly hypergraphs can be solved in polynomial time if p and q are fixed and co-NP-complete if p is part of the input. In addition, we generalize the characterization of p -clique-Helly graphs in terms of expansions to (p , q) -clique-Helly graphs and give different characterizations of hereditary (p , q) -clique-Helly graphs, including one by forbidden induced subgraphs. We give an improvement on the time bound for the recognition of (p , q) -clique-Helly graphs and prove that the recognition problem of hereditary (p , q) -clique-Helly graphs is polynomial-time solvable for p and q fixed and NP-hard if p or q is part of the input. Finally, we provide different characterizations, give recognition algorithms, and prove hardness results for (p , q) -biclique-Helly graphs and hereditary (p , q) -biclique-Helly graphs which are analogous to those for (p , q) -clique-Helly and hereditary (p , q) -clique-Helly graphs. [ABSTRACT FROM AUTHOR]

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]

A normal Helly circular-arc graph is the intersection graph of a set of arcs on a circle of which no three or less arcs cover the whole circle. Lin et al. (2013) characterized circular-arc graphs that are not normal Helly circular-arc graphs, and used them to develop the first recognition algorithm for this graph class. As open problems, they ask for the forbidden subgraph characterization and a direct recognition algorithm for normal Helly circular-arc graphs, both of which are resolved by the current paper. Moreover, when the input is not a normal Helly circular-arc graph, our recognition algorithm finds in linear time a minimal forbidden induced subgraph as a certificate. Our approach yields also a considerably simpler algorithm for the certifying recognition of proper Helly circular-arc graphs, a subclass of normal Helly circular-arc graphs. [ABSTRACT FROM AUTHOR]

A set of vertices X of a graph G is convex if no shortest path between two vertices in X contains a vertex outside X . We prove that for fixed p ≥ 1 , all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial time. [ABSTRACT FROM AUTHOR]