Solving NP-hard Problems on \textsc{GaTEx} Graphs: Linear-Time Algorithms for Perfect Orderings, Cliques, Colorings, and Independent Sets

TL;DR

The paper studies GaTEx graphs, a natural generalization of cographs, and develops linear-time algorithms for fundamental NP-hard problems on this class by leveraging galled-tree explanations and modular decomposition. By constructing prime-vertex replacement (pvr) networks that convert MD trees into galled-trees, the authors provide a unified framework to compute $\omega(G)$, $\chi(G)$, $\alpha(G)$, and to obtain perfect orderings. Central contributions include a linear-time perfect-ordering algorithm (Algorithm Sigma) and linear-time procedures for maximum cliques and independent sets, all grounded in the pvr-network structure. These results connect GaTEx graphs to various well-known graph classes and enable scalable combinatorial computation on this expressive graph family.

Abstract

The class of $\mathsf{Ga}$lled-$\mathsf{T}$ree $\mathsf{Ex}$plainable ($\mathsf{GaTEx}$) graphs has recently been discovered as a natural generalization of cographs. Cographs are precisely those graphs that can be uniquely represented by a rooted tree where the leaves correspond to the vertices of the graph. As a generalization, $\mathsf{GaTEx}$ graphs are precisely those that can be uniquely represented by a particular rooted acyclic network, called a galled-tree. This paper explores the use of galled-trees to solve combinatorial problems on $\mathsf{GaTEx}$ graphs that are, in general, NP-hard. We demonstrate that finding a maximum clique, an optimal vertex coloring, a perfect order, as well as a maximum independent set in $\mathsf{GaTEx}$ graphs can be efficiently done in linear time. The key idea behind the linear-time algorithms is to utilize the galled-trees that explain the $\mathsf{GaTEx}$ graphs as a guide for computing the respective cliques, colorings, perfect orders, or independent sets.

Figures (3)

• Figure 1: Shown is a GaTEx graph $G=(V,E)$ (top left) together with its modular decomposition tree $(\mathscr{T}_G,t_G)$ (top right) and a pvr-network $(N,t)$ (bottom right) that explains $G$. The graph $G$ has as strong modules the singletons $\{x\}$, $x\in V = \{a,b,c,\dots,g\}$, the entire vertex set $V$ and the sets $M_1 =\{d,e,f,g,h\}$ and $M_2=\{g,h\}$. Each vertex $w$ in $\mathscr{T}_G$ represent the strong module $L(\mathscr{T}_G(w))$. The graph $G$ has two prime modules, namely $M_1 = L(\mathscr{T}_G(v_2))$ and $V = L(\mathscr{T}_G(v_1))$. The respective quotient graphs $H_1 \coloneqq G/\mathbb{M}_{\max}(G)$ and $H_2 \coloneqq G[M_1]/\mathbb{M}_{\max}(G[M_1])$ are shown bottom left. The pvr-network $(N,t)$ is a galled-tree that is obtained from $(\mathscr{T}_G,t_G)$ by locally replacing the vertex $v_i$ by the strong quasi-discriminating elementary galled-tree $(N_{v_i},t_{v_i})$ that explains $H_i$, $i\in \{1,2\}$ (cf. Def. \ref{['def:pvr']}).
• Figure 2: Left a galled-tree $(N,t)$ that explains the GaTEx graph $G$ on the right. In addition, $G$ is equipped with a vertex coloring that is obtained with a greedy coloring based on the perfect order $cabdeghf$ computed with Algorithm \ref{['alg:sigma']}. Since $G[c,e,g,h]$ is a complete graph on four vertices, this coloring is optimal. see explanations in Example \ref{['exmpl:alg-order']} for further details.
• Figure 3: Left a galled-tree $(N,t)$ that explains the GaTEx graph $G$ on the right. Algorithm \ref{['alg:clique']} returns the induced subgraph $G[g,h,d,c]$ (highlighted in blue) which is a maximum clique of $G$. All vertices marked as active are highlighted with $\star$. Paths $P$ from $\rho_N$ to leaves $x$ in $N$ where all vertices $v\neq \rho_N$ in $P$ are active are highlighted in blue; see explanations in Example \ref{['exmpl:alg']} for further details.

Theorems & Definitions (35)

• Remark 2.1
• Proposition 2.2: CHVATAL198463
• Proposition 2.3: HS:22
• Definition 2.4: prime-vertex replacement (pvr) networks
• Lemma 3.1
• proof
• Example 3.2
• Proposition 3.3
• proof
• Proposition 3.4