The complexity of convexity number and percolation time in the cycle convexity

TL;DR

This work studies cycle convexity in graphs, detailing how the interval function $I(S) = S \cup \{u \mid G[S \cup {u}]$ has a cycle containing $u$\}$ governs the hull operator and associated parameters $con(G)$ and $pn(G)$. It establishes that deciding $con(G) \ge k$ is NP-complete and [1]-hard parameterized by $k$ on thick spiders, yet solvable in polynomial time for extended $P_4$-laden graphs; for percolation time, it shows NP-completeness for fixed $k \ge 9$ while giving polynomial-time algorithms for fixed $k \le 2$ and for cactus graphs. The authors provide a linear-time algorithm for cacti via a cycle-tree construction and a cubic-time method for fixed $k \le 2$, plus a perpetuation gadget-driven reduction from $3$-SAT to prove NP-hardness for $k \ge 9$, with extensions to larger fixed $k$. Overall, the paper delineates clear computational boundaries across graph classes and demonstrates decomposition-based approaches to compute $con(G)$ and $pn(G)$ in structured families such as extended $P_4$-laden graphs and cacti.

Abstract

The subject of graph convexity is well explored in the literature, the so-called interval convexities above all. In this work, we explore the cycle convexity, an interval convexity whose interval function is $I(S) = S \cup \{u \mid G[S \cup \{u\}]$ has a cycle containing $u\}$. In this convexity, we prove that determine whether the convexity number of a graph $G$ is at least $k$ is \NP-complete and \W[1]-hard when parameterized by the size of the solution when $G$ is a thick spider, but polynomial when $G$ is an extended $P_4$-laden graph. We also prove that determining whether the percolation time of a graph is at least $k$ is \NP-complete even for fixed $k \geq 9$, but polynomial for cacti or for fixed $k\leq2$.

Figures (11)

• Figure 1: Example of a quasi-spider obtained from an $(S,C,R)$-thick spider by replacing a vertex from $S$ by a $K_2$ with the vertices $v$ and $u$.
• Figure 2: Hierarchy of graphs with few $P_4$'s.
• Figure 3: An example of a hull set $S$ of a cactus, represented by the gray vertices. The vertices $v_1,v_2$ and $v_3$ and sequence of cycles $C_2,C_1,C_0$ are presented as defined.
• Figure 4: An example with the sequence of cycles $C_0,C_1,\ldots,C_k$, for $k = 2$, the vertices $v$ and $v'$, and the sets $S', L_G$ and $Q$ represented by the black vertices, vertices outlined in red and gray vertices, respectively.
• Figure 5: An example where $v$ has exactly one neighbor in $V(C') \setminus \{w\}$. The cycle $C$ is denoted by the blue edges, the cycle $C'$ by the red ones and $S$ by the gray vertices.

Theorems & Definitions (28)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Theorem 5: Giakoumakis giakoumakis1996
• Theorem 6