Counting Subgraphs in Somewhere Dense Graphs

TL;DR

This work analyzes the fixed-parameter tractability of counting subgraphs, induced subgraphs, and homomorphisms when both the pattern H and host G are drawn from restricted graph families. It develops a unified, hardness-driven framework that extends interpolation via graph tensoring to monotone somewhere-dense hosts through fractures and coloured subdivisions, yielding crisp dichotomies: #Match(H→G) and #IndSet(G) are FPT iff G is nowhere dense, with ETH-based lower bounds otherwise; for Hom, tractability hinges on nowhere-density and bounded pattern treewidth, with hardness emerging for somewhere-dense hosts and unbounded tw(H). It further provides exhaustive classifications for #Sub(H→G) and #IndSub(H→G) under monotone or hereditary assumptions, unifying and strengthening a broad spectrum of prior results (e.g., counting matchings in degenerate and bipartite graphs). The findings illuminate the structural boundaries between tractable and intractable counting problems in sparse-plus-structured graph classes, and pave the way for refined criteria (such as induced grid minors) in hereditary settings and broader pattern families. The results have potential implications for network analysis applications where both the pattern and host are restricted by natural sparsity or hereditary constraints.

Abstract

We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time $f(H)\cdot |G|^{O(1)}$ for some computable function $f$. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes $\mathcal{G}$ as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting $k$-matchings in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. (2) Counting $k$-independent sets in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if $\mathcal{G}$ is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting $k$-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in $F$-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting $k$-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).

Figures (3)

• Figure 1: A fractured graph $Q\raisebox{.2ex}{origin=c]{-15}{$\sharp$}}\sigma$ from PeyerimhoffRSSVW22. Left: a vertex $v \in V(Q)$ with incident edges $E_Q(v)=\{ ,,,,,\}$. Right: the splitting of $v$ in $Q\raisebox{.2ex}{origin=c]{-15}{$\sharp$}}\sigma$ for a fracture $\sigma$ where the partition $\sigma_v$ of $E_Q(v)$ consists of the two blocks $B_1 =\{ ,,\}$, and $B_2=\{,,\}$.
• Figure 2: The wall $W_{4,5}$ (left) and the grid $\boxplus_4$ (right).
• Figure 3: the tensor product of the $H^1$-coloured graphs $(G^1,c^1)$ and $(H^1\raisebox{.2ex}{origin=c]{-15}{$\sharp$}}\rho,c_\rho)$.

Theorems & Definitions (66)

• Theorem 1: Theorem 18.9 in Sparsity
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Definition 8
• Definition 10: Graph Invariants
• Theorem 11