Counting Subgraphs in Degenerate Graphs

Bera, Suman K.; Gishboliner, Lior; Levanzov, Yevgeny; Seshadhri, C.; Shapira, A.

doi:10.1145/3520240

Cited by 6 publications

References 43 publications

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Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Goldberg,

Roth

2023

Algorithmica

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Given a class of graphs $${\mathcal {H}}$$ H , the problem $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is defined as follows. The input is a graph $$H\in {\mathcal {H}}$$ H ∈ H together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes $${\mathcal {H}}$$ H the problem $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is fixed-parameter tractable (FPT), i.e., solvable in time $$f(|H|)\cdot |G|^{O(1)}$$ f ( | H | ) · | G | O ( 1 ) . Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is FPT if and only if the class of allowed patterns $${\mathcal {H}}$$ H is matching splittable, which means that for some fixed B, every $$H \in {\mathcal {H}}$$ H ∈ H can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes $${\mathcal {H}}$$ H , and (II) all tree pattern classes, i.e., all classes $${\mathcal {H}}$$ H such that every $$H\in {\mathcal {H}}$$ H ∈ H is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).

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Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Goldberg,

Roth

2023

Algorithmica

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Finding and counting small tournaments in large tournaments

Yuster

2025

Theoretical Computer Science

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Counting Subgraphs in Somewhere Dense Graphs

Bressan,

Ann Goldberg,

Meeks

et al. 2024

SIAM J. Comput.

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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) • |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 G as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis:Counting k-matchings in a graph G ∈ G is fixed-parameter tractable if and only if G is nowhere dense.Counting k-independent sets in a graph G ∈ G is fixed-parameter tractable if and only if G is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if 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).At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting.

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Counting Subgraphs in Degenerate Graphs

Cited by 6 publications

References 43 publications

Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Finding and counting small tournaments in large tournaments

Counting Subgraphs in Somewhere Dense Graphs

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