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Induced Minors and Region Intersection Graphs

Édouard Bonnet, Robert Hickingbotham

TL;DR

The paper addresses whether every $H$-induced-minor-free graph class can be realized as a region intersection graph over some $H'$-minor-free class. It provides a strong negative answer by constructing, for any positive integers $t$ and $g$, a graph that is $K_{6}^{(1)}$-induced-minor-free with girth at least $g$ but not belonging to $ ext{RIG}( ext{H})$ for any $H$ that is $K_t$-minor-free, using an extension of the Pohoata–Davies grid and the Graph Minor Structure Theorem. The argument shows a fundamental separation between induced-minor freeness and representability as RIGs over minor-free hosts, and it extends to arbitrarily large girth by incorporating higher subdivisions. This advances understanding of the limits of the RIG framework in capturing induced-minor-free classes and provides a concrete obstruction construction grounded in advanced minor theory.

Abstract

We show that for any positive integers $g$ and $t$, there is a $K_{6}^{(1)}$-induced-minor-free graph of girth at least $g$ that is not a region intersection graph over the class of $K_t$-minor-free graphs. This answers in a strong form the recently raised question of whether for every graph $H$ there is a graph $H'$ such that $H$-induced-minor-free graphs are region intersection graphs over $H'$-minor-free graphs.

Induced Minors and Region Intersection Graphs

TL;DR

The paper addresses whether every -induced-minor-free graph class can be realized as a region intersection graph over some -minor-free class. It provides a strong negative answer by constructing, for any positive integers and , a graph that is -induced-minor-free with girth at least but not belonging to for any that is -minor-free, using an extension of the Pohoata–Davies grid and the Graph Minor Structure Theorem. The argument shows a fundamental separation between induced-minor freeness and representability as RIGs over minor-free hosts, and it extends to arbitrarily large girth by incorporating higher subdivisions. This advances understanding of the limits of the RIG framework in capturing induced-minor-free classes and provides a concrete obstruction construction grounded in advanced minor theory.

Abstract

We show that for any positive integers and , there is a -induced-minor-free graph of girth at least that is not a region intersection graph over the class of -minor-free graphs. This answers in a strong form the recently raised question of whether for every graph there is a graph such that -induced-minor-free graphs are region intersection graphs over -minor-free graphs.
Paper Structure (7 sections, 8 theorems, 2 equations, 2 figures)

This paper contains 7 sections, 8 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard graph-theoretic notation
  4. Tree-decomposition
  5. Minors, induced minors, and region intersection graphs
  6. Graph minor structure theorem
  7. Proof of \ref{['thm:main']}

Key Result

Lemma 1

For every graph $G$, if a graph $H$ is not a minor of $G$ then any graph that contains $H^{(1)}$ as an induced minor is not a region intersection graph over $G$.

Figures (2)

  • Figure 1: The Pohoata--Davies $6 \times 6$ grid.
  • Figure 2: The graphs $B_n, B'_n$ and the order $\prec$. We only represented one entire copy of $A_n^{(1)}$. Black edges represent $B_n$. Together with the red edges, they form $B'_n$. Every vertex filled in gray is adjacent to the apex vertex to the right (we only drew some of these edges for legibility). The Hamiltonian path of $B'_n$ in blue defines the successor relation of $\prec$.

Theorems & Definitions (17)

  • Lemma 1: Lee17
  • Theorem 2
  • Theorem 3: robertson2003graph
  • Theorem 4: dujmovic2017layered
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Claim 1
  • ...and 7 more