Search

Let $G$ be a graph having a vertex $v$ such that $H = G - v$ is a trivially perfect graph. We give a polynomial-time algorithm for the problem of deciding whether it is possible to add at most $k$ edges to $G$ to obtain a trivially perfect graph. This is a slight variation of the well-studied {\sc Edge Completion}, also known as {\sc Minimum Fill-In}, problem. We also show that if $H$ is a chordal graph, then the problem of deciding whether it is possible to add at most $k$ edges to $G$ to obtain a chordal graph is \NP-complete., Comment: 23 pages, 1 figures

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 $(p,q)$-the 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. We prove several characterizations of hereditary $(p,q)$-Helly hypergraphs, including one by minimal forbidden partial subhypergraphs. 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 but co-NP-complete if $p$ is part of the input. In addition, we generalize to $(p,q)$-clique-Helly graphs the characterization of $p$-clique-Helly graphs in terms of expansions 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 but NP-hard if $p$ or $q$ is part of the input. Finally, we provide different characterizations, give recognition algorithms, and prove hardness results for (hereditary) $(p,q)$-biclique-Helly graphs., Comment: 28 pages

We present some complexity results concerning the problems of covering a graph with $p$ convex sets and of partitioning a graph into $p$ convex sets. The following convexities are considered: digital convexity, monophonic convexity, $P_3$-convexity, and $P_3^*$-convexity., Comment: 10 pages

A $(0,1)$-matrix has the Consecutive Ones Property (C1P) for the rows if there is a permutation of its columns such that the ones in each row appear consecutively. We say a $(0, 1)$-matrix is nested if it has the consecutive ones property for the rows (C1P) and every two rows are either disjoint or nested. We say a $(0, 1)$-matrix is 2-nested if it has the C1P and admits a partition of its rows into two sets such that the submatrix induced by each of these sets is nested. We say a split graph $G$ with split partition $(K, S)$ is nested (resp.\ 2-nested) if the matrix $A(S, K)$ which indicates the adjacency between vertices in $S$ and $K$ is nested (resp.\ 2-nested). In this work, we characterize nested and 2-nested matrices by minimal forbidden submatrices. This characterization leads to a minimal forbidden induced subgraph characterization for these classes of graphs, which are a superclass of threshold graphs and a subclass of split and circle graphs., Comment: 6 pages. Presented in the LatinAmerican Workshop on Cliques in Graphs 2018. Sent to publish in the special issue devoted to this conference of Mathematica Contempor\^anea

A normal Helly circular-arc graph is the intersection graph of arcs on a circle of which no three or less arcs cover the whole circle. Lin, Soulignac, and Szwarcfiter [Discrete Appl. Math. 2013] characterized circular-arc graphs that are not normal Helly circular-arc graphs, and used it to develop the first recognition algorithm for this graph class. As open problems, they ask for the forbidden induced 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 certificate., Comment: Preliminary results of this paper appeared in the proceedings of SBPO 2012 and FAW 2014

An interval graph is the intersection graph of a finite set of intervals on a line and a circular-arc graph is the intersection graph of a finite set of arcs on a circle. While a forbidden induced subgraph characterization of interval graphs was found fifty years ago, finding an analogous characterization for circular-arc graphs is a long-standing open problem. In this work, we study the intersection graphs of finite sets of arcs on a circle no three of which cover the circle, known as normal Helly circular-arc graphs. Those circular-arc graphs which are minimal forbidden induced subgraphs for the class of normal Helly circular-arc graphs were identified by Lin, Soulignac, and Szwarcfiter, who also posed the problem of determining the remaining minimal forbidden induced subgraphs. In this work, we solve their problem, obtaining the complete list of minimal forbidden induced subgraphs for the class of normal Helly circular-arc graphs., Comment: 15 pages, 3 figures. This paper originally appeared in proceedings of the XVI Congreso Latino-Iberoamericano de Investigaci\'on Operativa and the XLIV Simp\'osio Brasileiro de Pesquisa Operacional, September 24-28, 2012, Rio de Janeiro, Brazil. Anais do XLIV Simposio Brasileiro de Pesquisa Operacional, SOBRAPO, Rio de Janeiro, Brazil, 2012, pages 4093--4104. http://www.din.uem.br/sbpo/sbpo2012/pdf/arq0518.pdf

We initiate the study of graph classes of power-bounded clique-width, that is, graph classes for which there exist integers $k$ and $\ell$ such that the $k$-th powers of the graphs are of clique-width at most $\ell$. 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., Comment: 23 pages, 4 figures