Graph Classes Closed under Self-intersection

TL;DR

This paper examines finitely defined self-intersection-closed graph classes, an intermediate universe between monotone and hereditary classes, and establishes exact dichotomies for key problems. Central to the approach is a structural characterization of graphs in $\mathrm{Si\text{-}Free}(tS_{t,t,t})$ that exclude a tripod, showing that such graphs are either complete or have bounded treewidth, which enables polynomial-time solvability of Maximum (Weight) Independent Set and related problems. The authors extend these dichotomies to Maximum Induced Matching on bipartite graphs, and to Satisfiability/Counting Satisfiability on incidence graphs, as well as to the boundedness of clique-width for bipartite classes, all within finitely defined si-free families. The results generalize classical monotone-class dichotomies to the self-intersection setting, precisely delineating tractability boundaries and suggesting further research into infinite obstruction sets and broader graph parameters.

Abstract

A graph class is monotone if it is closed under taking subgraphs. It is known that a monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a so-called tripod, that is, a disjoint union of trees with exactly one vertex of degree 3 and paths. This dichotomy also characterizes exactly those monotone graph classes for which many NP-hard algorithmic problems admit polynomial-time algorithms. These algorithmic dichotomies, however, do not extend to the universe of all hereditary classes, which are classes closed under taking induced subgraphs. This leads to the natural question of whether we can extend known algorithmic dichotomies for monotone classes to larger families of hereditary classes. We give an affirmative answer to this question by considering the family of hereditary graph classes that are closed under self-intersection, which is known to be located strictly between the monotone and hereditary classes. We prove a new structural characterization of graphs in self-intersection-closed classes excluding a tripod. We use our characterization to give a complete dichotomy of Maximum Independent Set, and its weighted variant for self-intersection-closed classes defined by finitely many obstructions: these problems are in P if the class excludes a tripod and NP-hard otherwise. This generalizes several known results on Maximum Independent Set. We also use it to obtain dichotomies for Maximum Induced Matching on self-intersection-closed classes of bipartite graphs defined by finitely many obstructions. Similarly, we obtain dichotomies for Satisfiability and Counting Satisfiability on self-intersection-closed classes of (bipartite) incidence graphs defined by finitely many obstructions, and for boundedness of clique-width for self-intersection-closed classes of bipartite graphs defined by finitely many obstructions.

Figures (5)

• Figure 1: The graphs $X_3$ and $Y_3$.
• Figure 2: An illustration of the structure of the graph $H$ from \ref{['lem:clique-3-path']}. There are no edges between the vertices of $Q_1$ and $Q_2$, except for the edge $(y_1, y_2)$. This is indicated by highlighting these paths in blue and using thicker lines. In contrast, edges may exist between the vertices of $Q_3$ and those of $Q_1$ and $Q_2$.
• Figure 3: An illustration of the three cases in the proof of \ref{['th:clique-to-Sttt']}. Edges and paths highlighted in blue indicate that the their vertices induce a chordless cycle, thereby satisfying Assumption 7 of \ref{['lem:clique-3-path']}.
• Figure 4: An illustration of the proof of \ref{['lem:LtSttt-sic-tSttt']}. Both graphs $H$ (blue) and $H'$ (red) are isomorphic to $L(S_{4,4,4})$. Their intersection (highlighted vertices and thick double edges) is isomorphic to $S_{3,3,4}$.
• Figure 5: The graph $H_k$.

Theorems & Definitions (38)

• Lemma 1: Alekseev and Sorochan MR3812542
• Lemma 2
• proof
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Claim 5.1
• proof : Proof of \ref{['claim1']}
• Lemma 6