Counting Patterns in Degenerate Graphs in Constant Space

TL;DR

This paper studies counting problems for a fixed pattern graph $H$ in host graphs $G$ that are $d$-degenerate, focusing on subgraphs, induced subgraphs, and homomorphisms. It introduces DAG treedepth, a constant-space analogue of treedepth for directed acyclic graphs, and uses it to design constant-space divide-and-conquer algorithms for counting, contrasting it with the existing DAG treewidth framework. Key contributions include (i) constant-space algorithms for sparse patterns, (ii) an induced-minor-based obstruction characterization for $dtw\le 2$, (iii) $O(n^3)$ time counting for all nine-vertex patterns, (iv) improved runtimes for induced-subgraph counting with DAG treewidth, and (v) tight connections between DAG treedepth and treedepth through a source-graph family $\mathcal{G}$ and a $G_S$ construction. The work also shows that for patterns up to 11 vertices, $dtw\le 2$, enabling subquadratic or quadratic-time counting in bounded-degeneracy graphs, with conditional lower bounds for larger patterns. Overall, the results advance constant-space subgraph counting in sparse graphs and illuminate the trade-offs between treedepth- and treewidth-based approaches in this regime.

Abstract

For an arbitrary, fixed graph (pattern graph), we study the algorithmic complexity of counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms from the pattern graph to $n$-vertex, $d$-degenerate graphs as input. Recent work by Bressan (Algorithmica, 2021) has shown that this problem has efficient dynamic programming algorithms using a graph parameter called DAG treewidth. Bressan used DAG treewidth to design a fast algorithm for counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms that use polynomial space. Bera, Gishboliner, Levanzov, Seshadhri, and Shapira (SODA, 2021) provided a characterization of graphs with DAG treewidth one. In this paper, we introduce a new graph parameter called DAG treedepth and show that it yields efficient divide and conquer algorithms that use only constant space (in the unit-cost RAM model). Specifically, we show: An algorithm for counting subgraphs isomorphic to sparse pattern graphs using only constant space. We derive an induced minor-based characterization for graphs of DAG treedepth up to two. For pattern graphs upto nine vertices, the induced subgraphs can be counted in $O(n^3)$ time using constant space. An algorithm for counting induced subgraphs that matches the running time given by Bressan but only uses constant space. Apart from the DAG treedepth result, we also focus on DAG treewidth. For DAG treewidth, we show that we can count homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms faster than Bressan's algorithm (2021). We further show that for all pattern graphs up to 11 vertices, we can count induced subgraphs in quadratic time.

Figures (3)

• Figure 1: The black nodes are sources and the white nodes are non-sources. In figure (a), the DAG treedepth is 1, but the treedepth is $5$. In figure (b), DAG treedepth is 2, but treedepth is 4.
• Figure 2: $H_1$ and $H_2$ are obstruction for DAG treedepth 2.
• Figure 3: $k_4$ with one subdivision

Theorems & Definitions (82)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 5
• Theorem 6
• Theorem 7
• Conjecture 1
• Theorem 8
• Conjecture 2
• Theorem 9