Mader's Conjecture and Its Variants for Cographs

TL;DR

This paper proves Mader's conjecture and its variants for cographs, establishing conditions under which certain subtrees preserve connectivity properties, and provides tight bounds for these conditions.

Contribution

It extends Mader's conjecture to cographs, proving several variants and establishing tight bounds for subtree existence that preserve various connectivity levels.

Findings

Mader's conjecture holds for cographs with specific degree conditions.

Variants of Mader's conjecture are validated for cographs.

Tight bounds are provided for degree conditions ensuring connectivity-preserving subtrees.

Abstract

The class of cographs is one of the most well-known graph classes, which is also known to be equivalent to the class of $P_4$-free graphs. We show that Mader's conjecture is true if we restrict ourselves to cographs, that is, for any tree $T$ of order $m$, every $k$-connected cograph $G$ with $\delta(G) \geq \left\lfloor \frac{3k}{2} \right\rfloor +m-1$ contains a subtree $T' \cong T$ such that $G-V(T')$ is still $k$-connected, where $\delta(G)$ denotes the minimum degree of $G$. Moreover, we show that three variants of Mader's conjecture hold for cographs, that is, for any tree $T$ of order $m$, $\bullet$ every $k$-connected (respectively, $k$-edge-connected) cograph $G$ with $\delta(G) \geq k+m-1$ contains a subtree $T' \cong T$ such that $G-E(T')$ is $k$-connected (respectively, $k$-edge-connected), $\bullet$ every $k$-edge-connected cograph $G$ with $\delta(G) \geq k+m-[k = 1]$ contains a subtree $T' \cong T$ such that $G-V(T')$ is $k$-edge-connected, where we use Iverson's convention for $[k = 1]$. We furthermore present tight lower bounds on the minimum degree of a cograph for the existence of disjoint connectivity keeping trees, a maximal connectedness keeping tree and a super edge-connectedness keeping tree.