Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

On Minimum Maximal Distance-k Matchings

We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $\delta \ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum