Discrete Mathematics & Theoretical Computer Science |

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.

Section:
Special issues

We define a q-linear path in a hypergraph H as a sequence (e_1,...,e_L) of edges of H such that |e_i ∩ e_i+1 | ∈ [[1, q]] and e_i ∩ e_j = ∅ if |i − j| > 1. In this paper, we study the connected components associated to these paths when q = k − 2 where k is the rank of H. If k = 3 then q = 1 which coincides with the well-known notion of linear path or loose path. We describe the structure of the connected components, using an algorithmic proof which shows that the connected components can be computed in polynomial time. We then mention two consequences of our algorithmic result. The first one is that deciding the winner of the Maker-Breaker game on a hypergraph of rank 3 can be done in polynomial time. The second one is that tractable cases for the NP-complete problem of "Paths Avoiding Forbidden Pairs" in a graph can be deduced from the recognition of a special type of line graph of a hypergraph.

Section:
Special issues

A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected graphs and obtain a more detailed result that relates the number of vertices to the operations involved in constructing a respective uniformly $3$-connected graph. Furthermore, we investigate how crossing numbers and treewidths behave under the mentioned constructions. We demonstrate how these results can be utilized to study the structure and properties of uniformly $3$-connected graphs with minimum number of vertices of minimum degree.

Section:
Special issues

Inspired by the split decomposition of graphs and rank-width, we introduce the notion of $r$-splits. We focus on the family of $r$-splits of a graph of order $n$, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.

Section:
Special issues

This paper is devoted to the study of particular geometrically defined intersection classes of graphs. Those were previously studied by Magnant and Martin, who proved that these graphs have arbitrary large chromatic number, while being triangle-free. We give several structural properties of these graphs, and we raise several questions.

Section:
Special issues

Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where $\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of $\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$ such that each subset in $\mathcal{S}$ contains exactly one element in $\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as exactly hittable interval graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.

Section:
Special issues