Linear-Time Recognition of Double-Threshold Graphs

A graph $$G = (V,E)$$ G = ( V , E ) is a double-threshold graph if there exist a vertex-weight function $$w :V \rightarrow \mathbb {R}$$ w : V → R and two real numbers $$\mathtt {lb}, \mathtt {ub}\in \mathbb {R}$$ lb , ub ∈ R such that $$uv \in E$$ u v ∈ E if and only if $$\mathtt {lb}\le \mathtt {w}(u) + \mathtt {w}(v) \le \mathtt {ub}$$ lb ≤ w ( u ) + w ( v ) ≤ ub . In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in $$O(n^{3} m)$$ O ( n 3 m ) time, where n and m are the numbers of vertices and edges, respectively.

Representing Graphs as the Intersection of Cographs and Threshold Graphs

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\dim_{COG}(G)$ (resp. $\dim_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $\dim_{COG}(G)\leqslant\mathrm{tw}(G)+2$, (b) $\dim_{TH}(G)\leqslant\mathrm{pw}(G)+1$, and (c) $\dim_{TH}(G)\leqslant\chi(G)\cdot\mathrm{box}(G)$, where $\mathrm{tw}(G)$, $\mathrm{pw}(G)$, $\chi(G)$ and $\mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $\dim_{COG}(G)$ and $\dim_{TH}(G)$ when $G$ belongs to some special graph classes.

Relating centralities in graphs and the principal eigenvector of its distance matrix

In this work a new centrality measure of graphs is presented, based on the principal eigenvector of the distance matrix: spectral closeness. Using spectral graph theory, we show some of its properties and we compare the results of this new centrality with closeness centrality. In particular, we prove that for threshold graphs these two centralities always coincide. In addition we construct an infinity family of graphs for which these centralities never coincide.