On bounding the difference between the maximum degree and the chromatic number by a constant

We provide a finite forbidden induced subgraph characterization for the graph class $\varUpsilon_k$, for all $k \in \mathbb{N}_0$, which is defined as follows. A graph is in $\varUpsilon_k$ if for any induced subgraph, $\Delta \leq \chi -1 + k$ holds, where $\Delta$ is the maximum degree and $\chi$ is the chromatic number of the subgraph. We compare these results with those given in [O. Schaudt, V. Weil, On bounding the difference between the maximum degree and the clique number, Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI: 10.1007/s00373-014-1468-3], where we studied the graph class $\varOmega_k$, for $k \in \mathbb{N}_0$, whose graphs are such that for any induced subgraph, $\Delta \leq \omega -1 + k$ holds, where $\omega$ denotes the clique number of a graph. In particular, we give a characterization in terms of $\varOmega_k$ and $\varUpsilon_k$ of those graphs where the neighborhood of every vertex is perfect.Comment: 10 pages, 4 figure

A Graph Class related to the Structural Domination Problem

In the structural domination problem one is concerned with the question if a given graph has a connected dominating set whose induced subgraph has certain structural properties. For most of the common graph properties, the associated decision problem is NP-hard. Recently, Bacsô and Tuza gave a full characterization of the graphs whose every induced subgraph has a connected dominating set satisfying an arbitrary prescribed hereditary property. Using the Theorem of Bacsô and Tuza, we derive a finite forbidden subgraph characterization of the distance-hereditary graphs that have a dominating induced tree. Furthermore, we show that for distance-hereditary graphs minimum dominating induced trees can be found efficiently. The main part of the paper studies a new class of graphs, the extit{structural domination class}, which is closely related to the structural domination problem. Among other results, we give new characterizations of certain subclasses of distance-hereditary graphs (in particular for ptolemaic graphs) and analyse the structure of minimum connected dominating sets of structural domination graphs. It turns out that many of the problems associated to structural domination become tractable on the hereditary part of the structural domination class

A Graph Class related to the Structural Domination Problem

In the structural domination problem one is concerned with the question if a given graph has a connected dominating set whose induced subgraph has certain structural properties. For most of the common graph properties, the associated decision problem is NP-hard. Recently, Bacsô and Tuza gave a full characterization of the graphs whose every induced subgraph has a connected dominating set satisfying an arbitrary prescribed hereditary property. Using the Theorem of Bacsô and Tuza, we derive a finite forbidden subgraph characterization of the distance-hereditary graphs that have a dominating induced tree. Furthermore, we show that for distance-hereditary graphs minimum dominating induced trees can be found efficiently. The main part of the paper studies a new class of graphs, the extit{structural domination class}, which is closely related to the structural domination problem. Among other results, we give new characterizations of certain subclasses of distance-hereditary graphs (in particular for ptolemaic graphs) and analyse the structure of minimum connected dominating sets of structural domination graphs. It turns out that many of the problems associated to structural domination become tractable on the hereditary part of the structural domination class

Efficient total domination in digraphs

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X . Not every digraph has an efficiently total dominating set. We study graphs that permit an orientation having such a set and give complexity results and characterizations concerning this question. Furthermore, we study the computational complexity of the (weighted) efficient total domination problem for several digraph classes. In particular we deal with most of the common generalizations of tournaments, like locally semicomplete and arc-locally semicomplete digraphs