Mader's Conjecture and Its Variants for Cographs

TL;DR

The paper proves Mader's connectivity-preserving-tree conjecture for the entire class of cographs ($P_4$-free graphs) and derives several variants for both $k$-connected and $k$-edge-connected cographs. It leverages the cotree/cocomponent structure to obtain constructive, case-based minimum-degree bounds that guarantee the existence of a subtree $T' \cong T$ with the residual graph preserving the desired connectivity. The results include tight lower bounds, characterizations of maximally connected and super edge-connected cographs, and algorithms for finding the required trees. Collectively, this work establishes Mader-type connectivity preserving trees for all $k\ge1$ within cographs and lays groundwork for extensions to broader graph classes.

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 $δ(G) \geq \left\lfloor \frac{3k}{2} \right\rfloor +m-1$ contains a subtree $T' \cong T$ such that $G-V(T')$ is still $k$-connected, where $δ(G)$ 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 $δ(G) \geq k+m-1$ 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 $δ(G) \geq k+m-[k = 1]$ 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.

Figures (4)

• Figure 1: A cograph $G$ and its parse tree.
• Figure 2: The cocomponents $G_1, G_2, G_3$ of the cograph $G$ in Fig. 1 and the cotree of $G$, where $G_1 = \langle \{a,b,c,d\} \rangle_G$, $G_2 = \langle \{e,f\} \rangle_G$ and $G_3 = \langle \{g\} \rangle_G$.
• Figure 3: The cardinalities of $V_1,V_2, V(G_1)$ and $S$ with the values $k,m,p,q$ and $r$ in Case 3.2 in the proof of Theorem \ref{['max-con-tree']}.
• Figure 4: The connected cographs of order at most 4.

Theorems & Definitions (49)

• Conjecture 1
• Theorem 1
• Proposition 1
• Conjecture 2
• Theorem 2
• Theorem 3
• Theorem 4
• Conjecture 3
• Lemma 1
• Lemma 2