Ideally Connected Cographs and Chordal Graphs

TL;DR

The paper characterizes ideal connectedness within two fundamental graph classes. It establishes that cographs are ideally connected exactly when they are $2K_2$-free, and that chordal graphs are ideally connected precisely when they are threshold graphs, linking ideal connectivity to well-studied graph constructions. A general structure theorem for ideally connected graphs with a $\kappa$-clique cut $S$ is developed, describing how $S$-subgraphs glue together and which degree conditions must hold. This framework is then specialized to chordal graphs, yielding a crisp threshold-graph characterization, and it is used to reveal a universal clique-tree property for threshold graphs, offering insights into their clique-structure representations.

Abstract

For distinct vertices $u,v$ in a graph $G$, let $κ_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then, $κ_G(u,v) \leq \min\{ \mbox{deg}_G(u), \mbox{deg}_G(v) \}$. If equality is attained for every pair of vertices in $G$, then $G$ is called ideally connected. In this paper, we characterize the ideally connected graphs in two well-known graph classes: the cographs and the chordal graphs. We show that the ideally connected cographs are precisely the $2K_2$-free cographs, and the ideally connected chordal graphs are precisely the threshold graphs, the graphs that can be constructed from the single-vertex graph by repeatedly adding either an isolated vertex or a dominating vertex.

Figures (4)

• Figure 1: A threshold graph with sixteen vertices.
• Figure 2: An example showing how an ideally connected graph with a $\kappa$-clique cut $S$ is obtained by gluing together smaller ideally connected graphs along $S$. On the left is an ideally connected graph $G$ with a $\kappa$-clique cut $S = \{ s_1, s_2\}$. On the right, we see how $G$ arises by gluing together ideally connected graphs along a clique of size two, according to Theorem \ref{['thm: conditions']}. The graphs $H_1, \ldots, H_4$ are the $S$-subgraphs of $G$, and $H_1$ is ideally connected and of connectivity at least two while $H_2$, $H_3$, and $H_4$ are $2$-regular and ideally connected, so are cycles.
• Figure 3: A threshold graph with five maximal cliques.
• Figure 4: A split graph that is not a threshold graph and is not clique tree universal.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm: intro cograph']}
• Corollary 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof