On the Kernel and Related Problems in Interval Digraphs

Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to be an independent set if no two vertices in $S$ are adjacent in $G$. A kernel (resp. solution) of $G$ is an independent and absorbing (resp. dominating) set in $G$. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph $G$ is an interval digraph if a pair of intervals $(S_u,T_u)$ can be assigned to each vertex $u$ of $G$ such that $(u,v)\in E(G)$ if and only if $S_u\cap T_v\neq\emptyset$. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs -- which arise when we require that the two intervals assigned to a vertex have to intersect. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs.Comment: 26 pages, 3 figure

Efficient Algorithms for Graphs with Few P-4’s

We show that a large variety of NP-complete problems can be solved efficiently for graphs with \u27few\u27 P4\u27s. We consider domination problems (domination, total domination, independent domination. connected domination and dominating clique), the Steiner tree problem, the vertex ranking problem, the pathwidth problem, the path cover number problem, the hamiltonian circuit problem, the list coloring problem and the precoloring extension problem. We show that all these problems can be solved in linear time for the class of (q,q - 4)-graphs, for every fixed q. These are graphs for which no set of at most q. vertices induces more than q - 4 different P4\u27s. © 2001 Elsevier Science B.V. All rights reserved

Dominating cliques in graphs

AbstractA set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding a dominating clique. A forbidden subgraph characterization is given for a class of graphs that have a connected dominating set of size three

Upper clique transversals in graphs

A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the study of the "upper" variant of this parameter, the upper clique transversal number, defined as the maximum size of a minimal clique transversal. We investigate this parameter from the algorithmic and complexity points of view, with a focus on various graph classes. We show that the corresponding decision problem is NP-complete in the classes of chordal graphs, chordal bipartite graphs, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs and proper interval graphs.Comment: Full version of a WG 2023 pape