On graphs with a simple structure of maximal cliques

TL;DR

This work introduces clique-sparse graph classes, defined by bounded clique-incidence-degree and clique-incidence-diversity, and provides several equivalent characterisations including a forbidden-induced-subgraph description. It develops two main tools—the clique-quotient graph and induced width parameters—to lift Menger’s theorem and the Grid Theorem to induced settings, under refined measures such as theta-treewidth and alpha-treewidth. The authors establish approximate induced Menger and induced grid theorems for clique-sparse graphs, relate these parameters to rankwidth, and show a finite obstruction set governs the clique-sparse property. Overall, the paper offers a unified framework connecting induced-structure results with classical width concepts, clarifying when rankwidth and alpha-treewidth align in restricted graph classes and highlighting the role of true-twins and clique interactions.

Abstract

We say that a hereditary graph class $\mathcal{G}$ is \emph{clique-sparse} if there is a constant $k=k(\mathcal{G})$ such that for every graph $G\in\mathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal clique of $G$ can be intersected in at most $k$ different ways by other maximal cliques. We provide various characterisations of clique-sparse graph classes, including a list of five parametric forbidden induced subgraphs. We show that recent techniques for proving induced analogues of Menger's Theorem and the Grid Theorem of Robertson and Seymour can be lifted to prove induced variants in clique-sparse graph classes when replacing ``treewidth'' by ''tree-independence number''.

Figures (4)

• Figure 1: The graph $G$ has four maximal cliques $\mathcal{K}(G) = \Set{\textcolor{K1colour}{K_1},\textcolor{K2colour}{K_2},\textcolor{K3colour}{K_3},\textcolor{K4colour}{K_4}}$. On the left side of the illustration, we see the linegraph $\mathrm{I}(\mathcal{K}(G))$. On the right we see the quotient graph $\widetilde{G}$ of $G$. The coloured vertices in $\widetilde{G}$ represent contracted classes of true twins. Note that ${\Delta_{\mathcal{K}}}(G) =3$, $\widetilde{\omega}(G) = 4$, and ${\mathsf{cdeg}}(G) = 3$.
• Figure 2: The forbidden induced subgraphs for clique-sparse graphs: the filled ellipses represent cliques, while the empty ellipses represent independent sets. See \ref{['sec:Prelim']} for precise definitions.
• Figure 3: From left to right, the graphs $\mathscr{M}^{\hbox{$\Leftcircle\mkern-12mu\Rightcircle$}}_5$, $\mathscr{M}^{\hbox{$\LEFTCIRCLE\mkern-12mu\Rightcircle$}}_5$, $\mathscr{M}^{\hbox{$\LEFTCIRCLE\mkern-12mu\RIGHTCIRCLE$}}_5$, $\mathscr{A}^{\hbox{$\Leftcircle\mkern-12mu\Rightcircle$}}_5$, $\mathscr{A}^{\hbox{$\LEFTCIRCLE\mkern-12mu\Rightcircle$}}_5$, $\mathscr{A}^{\hbox{$\LEFTCIRCLE\mkern-12mu\RIGHTCIRCLE$}}_5$, $\mathscr{H}^{\hbox{$\Leftcircle\mkern-12mu\Rightcircle$}}_5$, $\mathscr{H}^{\hbox{$\LEFTCIRCLE\mkern-12mu\Rightcircle$}}_5$, $\mathscr{H}^{\hbox{$\LEFTCIRCLE\mkern-12mu\RIGHTCIRCLE$}}_5$. The filled ellipses represent cliques, while the empty ellipses represent independent sets.
• Figure 4: Top row: $Q_5(\mathscr{M}^{\hbox{$\Leftcircle\mkern-12mu\Rightcircle$}}_5)$, $Q_5(\mathscr{M}^{\hbox{$\LEFTCIRCLE\mkern-12mu\Rightcircle$}}_5)$, and $Q_5(\mathscr{M}^{\hbox{$\LEFTCIRCLE\mkern-12mu\RIGHTCIRCLE$}}_5)$. Bottom row: $Q_5(\mathscr{H}^{\hbox{$\Leftcircle\mkern-12mu\Rightcircle$}}_5)$, $Q_5(\mathscr{H}^{\hbox{$\LEFTCIRCLE\mkern-12mu\RIGHTCIRCLE$}}_5)$, and $Q_5(\mathscr{A}^{\hbox{$\LEFTCIRCLE\mkern-12mu\RIGHTCIRCLE$}}_5)$.

Theorems & Definitions (50)

• Theorem 1.1: Gartland, Korhonen, Lokshtanov 2023$^{+}$ GartlandKL2023-inducedMengerMaxDeg
• Theorem 1.1
• Theorem 1.2: Korhonen, 2023 Korhonen2023-inducedgridmaxdegree
• Theorem 1.2
• Theorem 1.3
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• proof