A new approach on locally checkable problems

By providing a new framework, we extend previous results on locally checkable problems in bounded treewidth graphs. As a consequence, we show how to solve, in polynomial time for bounded treewidth graphs, double Roman domination and Grundy domination, among other problems for which no such algorithm was previously known. Moreover, by proving that fixed powers of bounded degree and bounded treewidth graphs are also bounded degree and bounded treewidth graphs, we can enlarge the family of problems that can be solved in polynomial time for these graph classes, including distance coloring problems and distance domination problems (for bounded distances)

Characterising circular-arc contact B0B_0-VPG graphs

A contact $B_0$-VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown by Deniz et al. that Recognition is $\mathsf{NP}$-complete for contact $B_0$-VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact $B_0$-VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs

On some special classes of contact B0B_0-VPG graphs

A graph $G$ is a $B_0$-VPG graph if one can associate a path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect at at least one grid-point. A graph $G$ is a contact $B_0$-VPG graph if it is a $B_0$-VPG graph admitting a representation with no two paths crossing and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact $B_0$-VPG graphs within four special graph classes: chordal graphs, tree-cographs, $P_4$-tidy graphs and $P_5$-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact $B_0$-VPG graphs.Comment: 34 pages, 15 figure

Forbidden induced subgraph characterization of circle graphs within split graphs

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there are diverse characterizations of circle graphs, a structural characterization by minimal forbidden induced subgraphs for the entire class of circle graphs is not known, not even restricted to split graphs (which are the graphs whose vertex set can be partitioned into a clique and a stable set). In this work, we give a characterization by minimal forbidden induced subgraphs of circle graphs, restricted to split graphs.Comment: 59 pages, 15 figure

Precedence thinness in graphs

Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of $k$ interval graphs, by splitting the graph in some special way. As a recent example of such an approach, the classes of $k$-thin and proper $k$-thin graphs have been introduced generalizing interval and proper interval graphs, respectively. The complexity of the recognition of each of these classes is still open, even for fixed $k \geq 2$. In this work, we introduce a subclass of $k$-thin graphs (resp. proper $k$-thin graphs), called precedence $k$-thin graphs (resp. precedence proper $k$-thin graphs). Concerning partitioned precedence $k$-thin graphs, we present a polynomial time recognition algorithm based on $PQ$-trees. With respect to partitioned precedence proper $k$-thin graphs, we prove that the related recognition problem is \NP-complete for an arbitrary $k$ and polynomial-time solvable when $k$ is fixed. Moreover, we present a characterization for these classes based on threshold graphs.Comment: 33 page

Thinness of product graphs

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function.Comment: 45 page

Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. Proper thinness is defined analogously, generalizing proper interval graphs, and a larger family of NP-complete problems are known to be polynomially solvable for graphs with bounded proper thinness. It is known that the thinness of a graph is at most its pathwidth plus one. In this work, we prove that the proper thinness of a graph is at most its bandwidth, for graphs with at least one edge. It is also known that boxicity is a lower bound for the thinness. The main results of this work are characterizations of 2-thin and 2-proper thin graphs as intersection graphs of rectangles in the plane with sides parallel to the Cartesian axes and other specific conditions. We also bound the bend number of graphs with low thinness as vertex intersection graphs of paths on a grid ($B_k$-VPG graphs are the graphs that have a representation in which each path has at most $k$ bends). We show that 2-thin graphs are a subclass of $B_1$-VPG graphs and, moreover, of monotone L-graphs, and that 3-thin graphs are a subclass of $B_3$-VPG graphs. We also show that $B_0$-VPG graphs may have arbitrarily large thinness, and that not every 4-thin graph is a VPG graph. Finally, we characterize 2-thin graphs by a set of forbidden patterns for a vertex order.Comment: An extended abstract of this work, entitled "Intersection models for 2-thin and proper 2-thin graphs", was accepted for LAGOS 2021 and will appear in ENTC

On some graph classes related to perfect graphs: A survey

International audiencePerfect graphs form a well-known class of graphs introduced by Berge in the 1960s in terms of a min-max type equality involving two famous graph parameters. In this work, we study variants and subclasses of perfect graphs defined by means of min-max relations of other graph parameters. Our focus is on clique-perfect, coordinated, and neighborhoodperfect graphs. We show the connection between graph classes and both hypergraph theory and the clique graph operator. We review different partial characterizations of them by forbidden induced subgraphs, present the previous results, and the main open problems. Computational complexity problems are also discussed

On some special classes of contact B₀-VPG graphs

A graph G is a B0-VPG graph if one can associate a horizontal or vertical path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid-point. A graph G is a contact B0-VPG graph if it is a B0-VPG graph admitting a representation with no one-point paths, no two paths crossing, and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact B0-VPG graphs within four special graph classes: chordal graphs, tree-cographs, P4-tidy graphs and P5-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact B0-VPG graphs.Centro de Investigación de Matemátic