A Complexity Analysis of the c-Closed Vertex Deletion Problem

TL;DR

This work investigates $c$-Closed Vertex Deletion ($c$-CVD), the problem of deleting at most $k$ vertices to obtain a $c$-closed graph. It presents a spectrum of results: NP-hardness for restricted graphs (e.g., bipartite graphs with bounded degree), kernel lower bounds and constructive upper bounds (including a kernel of size $O(k^{c+2})$ and a $O(x^3+x^2 c)$-sized kernel parameterized by $x$, the number of vertices in bad pairs), and positive results such as a linear-time algorithm for unit interval graphs with depth $ ext{at most } c+1$ and FPT results with respect to neighborhood diversity. The paper also connects closure concepts to weak closure and demonstrates how the new bad-pair parameter can yield compact kernels and tractable cases. Overall, it delineates hardness and tractability boundaries for $c$-CVD and suggests fruitful directions for further study via targeted graph classes and parameters.

Abstract

A graph is $c$-closed when every pair of nonadjacent vertices has at most $c-1$ common neighbors. In $c$-Closed Vertex Deletion, the input is a graph $G$ and an integer $k$ and we ask whether $G$ can be transformed into a $c$-closed graph by deleting at most $k$ vertices. We study the classic and parameterized complexity of $c$-Closed Vertex Deletion. We obtain, for example, NP-hardness for the case that $G$ is bipartite with bounded maximum degree. We also show upper and lower bounds on the size of problem kernels for the parameter $k$ and introduce a new parameter, the number $x$ of vertices in bad pairs, for which we show a problem kernel of size $\mathcal{O}(x^3 + x^2\cdot c))$. Here, a pair of nonadjacent vertices is bad if they have at least $c$ common neighbors. Finally, we show that $c$-Closed Vertex Deletion can be solved in polynomial time on unit interval graphs with depth at most $c+1$ and that it is fixed-parameter tractable with respect to the neighborhood diversity of $G$.

Figures (6)

• Figure 1: Hardness for different values of $c$ and $\Delta$. Red means NP-hard, green means polynomial-time solvable, and gray means unknown.
• Figure 2: Minimal forbidden subgraphs (FSG) for $c = 2,3,4$. The grey vertices are the bad pair, the black vertices are the connecting vertices, the full edges are the critical edges, and the dashed edges are optional edges.
• Figure 3: Construction of a $4$-CVD instance with maximum degree $6$ from a Vertex Cover instance with maximum degree $3$. The vertices from the Vertex Cover instance are black, the bad pair vertices are white and the additional connecting vertices are gray.
• Figure 4: Construction of a $3$-CVD instance with maximum degree $5$ from a Vertex Cover instance with maximum degree $3$. The vertices from the Vertex Cover instance are black. The gray square vertices are the connecting vertices for the FSGs corresponding to the edges incident with the degree-$3$ vertex. The white circle vertices are the bad pair vertices for the FSGs corresponding to other edges with the gray circle vertices as additional connecting vertices.
• Figure 5: Connected components for max degree $3$ graphs with $c = 3$ after exhaustively applying Rule \ref{['rrule:non_critical_edges']}. The black vertices are the bad pair in the initial FSG. The gray vertices are the connecting vertices of the bad pair and the white vertices are additional vertices that may or may not be part of the component.

Theorems & Definitions (36)

• proposition thmcounterproposition
• proof
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof : for $c = \Delta = 2$