Computing Subset Vertex Covers in $H$-Free Graphs

TL;DR

Subset Vertex Cover generalizes Vertex Cover by asking for a small set $S$ that covers all edges incident to a subset $T$ of vertices. The authors develop both hardness and tractability results: NP-completeness on specific sparse classes (subcubic planar claw+diamond-free and 2-unipolar graphs) and a set of polynomial-time dichotomies for $H$-free inputs, including $(sP_1+P_2+P_3)$-free and bounded mim-width graphs. They leverage reductions to Vertex Cover on probe graphs, along with combinatorial tools like neighbor equivalence and representative sets, to design efficient algorithms. The work yields a substantial partial classification for Subset Vertex Cover on $H$-free graphs and highlights cases where the subset variant is strictly harder than the classical Vertex Cover problem.

Abstract

We consider a natural generalization of Vertex Cover: the Subset Vertex Cover problem, which is to decide for a graph $G=(V,E)$, a subset $T\subseteq V$ and integer $k$, if $V$ has a subset $S$ of size at most $k$, such that $S$ contains at least one end-vertex of every edge incident to a vertex of $T$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We solve two open problems from the literature by proving that Subset Vertex Cover is NP-complete on subcubic (claw,diamond)-free planar graphs and on $2$-unipolar graphs, a subclass of $2P_3$-free weakly chordal graphs. Our results show for the first time that Subset Vertex Cover is computationally harder than Vertex Cover (under P $\neq$ NP). We also prove new polynomial time results, some of which follow from a reduction to Vertex Cover restricted to classes of probe graphs. We first give a dichotomy on graphs where $G[T]$ is $H$-free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs $G$, for which $G[T]$ is $H$-free, if $H=sP_1+tP_2$ and NP-complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for $(sP_1+P_2+P_3)$-free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for Subset Vertex Cover on $H$-free graphs.

Figures (6)

• Figure 1: An instance $(G,T,k)$ of Subset Vertex Cover, where $T$ consists of the orange vertices, together with a solution $S$ (a $T$-vertex cover of size $5$). Note that $S$ consists of four vertices of $T$ and one vertex of $\overline{T}=V\setminus T$.
• Figure 2: The graph $G'$ from Theorem \ref{['t-hard1']}, where $T = V\setminus W$ consists of the orange vertices.
• Figure 3: The graph $G'$ from Theorem \ref{['t-hard2']}, where the orange vertices form $T = V\setminus V(G^*)$.
• Figure 4: An example of the $2P_2$-free graph $G'$ of the proof of Theorem \ref{['t-sp2']}. Here, $T$ consists of the orange vertices. A solution $S$ can be split up into a minimal vertex cover $R$ of $G'[T]$ and a vertex cover $W$ of $G[V\setminus R]$.
• Figure 5: An illustration of the graph $G$ in the proof of Case 2 of Theorem \ref{['t-sp1p2p3']}, where $T$ consists of the orange vertices, and $p=3$. Edges in $G[V\setminus T]$ are not drawn, and for $x_2$ and $x_3$ some edges are partially drawn. Vertices $x_1, x_4, x_5,u,v$ do not belong to $W$, as they do not satisfy one of the Properties 1--3, whereas $x_2$ belongs to $W$, as $x_2$ satisfies Property 1 for $D_2$ (and also Property 2 for $D_2$ and $D_3$), and $x_3$ belongs to $W$, as $x_3$ satisfies Property 3 for $D_3$.

Theorems & Definitions (46)

• Proposition 1
• Theorem 2: Al82
• Theorem 3: BJPP22
• Theorem 4: GL04GLS84
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 9
• proof