Proportionally dense subgraphs of maximum size in degree-constrained graphs

TL;DR

Proportionally Dense Subgraphs (PDS) generalize community-like structure by requiring each vertex in a subgraph to have at least as large a proportion of neighbors inside as outside. This work maps the complexity of MaxPDS and connected MaxPDS across density-related graph parameters, including the maximum degree $Δ$, the $h$-index, and degeneracy $degen$, as well as parameters of the complement graph. It establishes para-NP-hardness for MaxPDS when parameterized by $(Δ, degen)$ and proves NP-hardness for dense-graph instances via $Δ(ar{G})=6$ and $degen(ar{G})=2$ with a bipartite complement, while providing polynomial-time algorithms for cases with $h≤2$ and for graphs with $h(ar{G})≤2$ (equivalently $Δ(ar{G})≤2$). The results extend to connected MaxPDS, and the paper includes a summary table and several constructive reductions to illuminate the boundary between intractable and tractable instances. These findings advance theoretical understanding of how global density interacts with local proportionality in graph partitioning problems and inform practical approaches for community detection under constrained density regimes.

Abstract

A proportionally dense subgraph (PDS) of a graph is an induced subgraph of size at least two such that every vertex in the subgraph has proportionally as many neighbors inside as outside of the subgraph. Then, maxPDS is the problem of determining a PDS of maximum size in a given graph. If we further require that a PDS induces a connected subgraph, we refer to such problem as connected maxPDS. In this paper, we study the complexity of maxPDS with respect to parameters representing the density of a graph and its complement. We consider $Δ$, representing the maximum degree, $h$, representing the $h$-index, and degen, representing the degeneracy of a graph. We show that maxPDS is NP-hard parameterized by $Δ,h$ and degen. More specifically, we show that maxPDS is NP-hard on graphs with $Δ=4$, $h=4$ and degen=2. Then, we show that maxPDS is NP-hard when restricted to dense graphs, more specifically graphs $G$ such that $Δ(\overline{G})\leq 6$, and graphs $G$ such that $degen(\overline{G}) \leq 2$ and $\overline{G}$ is bipartite, where $\overline{G}$ represents the complement of $G$. On the other hand, we show that maxPDS is polynomial-time solvable on graphs with $h\le2$. Finally, we consider graphs $G$ such that $h(\overline{G})\le 2$ and show that there exists a polynomial-time algorithm for finding a PDS of maximum size in such graphs. This result implies polynomial-time complexity on graphs with $n$ vertices of minimum degree $n-3$, i.e. graphs $G$ such that $Δ(\overline{G})\le 2$. For each result presented in this paper, we consider connected maxPDS and explain how to extend it when we require connectivity.

Figures (6)

• Figure 1: On the left we present an example of a cubic graph $G$ on $8$ vertices. On the right, we present the construction explained above. A double arrow $X \overset{x}{\underset{y}{\rightleftharpoons}} Y$ represents that the vertices in $X$ have at most $x$ neighbors in $Y$ and the vertices in $Y$ have at most $y$ neighbors in $X$.
• Figure 2: The adjacency between the sets $V_A$ and $V_E$ (resp. $V_B$ and $V_G$). The $\{0,1\}$-labeling represent a 2-coloring of $G'$.
• Figure 3: On the left we present an example of a cubic graph $G$ on $8$ vertices. On the right, we present the complement of the graph $G'$ constructed as explained above.
• Figure 4: The graph $G'$.
• Figure 5: The graph $G'$. Note that the green vertices correspond to the vertices $a^*$ and $b^*$.

Theorems & Definitions (45)

• Definition 1
• Definition 2
• Theorem 1
• Lemma 1
• proof
• Corollary 1
• proof
• Lemma 2
• proof
• Theorem 2