Parameterized complexity of the f-Critical Set problem

TL;DR

The paper studies the $f$-reversible process on graphs and the associated $f$-critical sets, formalizing the decision problem $f$-Critical Set and the critical-set number $r^c_f(G)$. It establishes a nuanced parameterized complexity landscape: NP-complete even for planar subcubic bipartite graphs with $m(f)\le 2$, and $W[1]$-hard for treewidth; meanwhile, $r^c_f(G)\le k$ is $\FPT$ for combined parameters $tw(G)+m(f)$, $tw(G)+\Delta(G)$, and for $k$ itself, with kernelizations for certain parameter blends and polynomial kernels of sizes $O(k\cdot m(f))$ and $O(k\cdot \Delta(G))$. The authors design an $\FPT$ algorithm with respect to $k$ based on a decomposition into forced-in/forced-out/may-in vertices and a bounded search over connected subsets, and show that the reversible model can be more tractable than the irreversible counterpart in some regimes. The results delineate a clear boundary between tractability and hardness and open avenues for kernels, sparse graph analysis, and approximation in the reversible setting.

Abstract

Given a graph $G=(V,E)$ and a function $f:V(G) \rightarrow \mathbb{N}$, an $f$-reversible process on $G$ is a dynamical system such that, given an initial vertex labeling $c_0 : V(G) \rightarrow \{0,1\}$, every vertex $v$ changes its label if and only if it has at least $f(v)$ neighbors with the opposite label. The updates occur synchronously in discrete time steps $t=0,1,2,\ldots$. An $f$-critical set of $G$ is a subset of vertices of $G$ whose initial label is $1$ such that, in an $f$-reversible process on $G$, all vertices reach label $1$ within one time step and then remain unchanged. The critical set number $r^c_f(G)$ is the minimum size of an $f$-critical set of $G$. Given a graph $G$, a threshold function $f$, and an integer $k$, the $f$-Critical Set problem asks whether $r^c_f(G) \leq k$. We prove that this problem is NP-complete for planar subcubic bipartite graphs and $m(f) \leq 2$, where $m(f)$ is the largest value of $f(v)$ over all $v \in V(G)$, and is W[1]-hard when parameterized by the treewidth $tw(G)$ of $G$. Additionally, we show that the problem is in FPT when parameterized by $tw(G)+m(f)$, $tw(G)+Δ(G)$, and $k$, where $Δ(G)$ denotes the maximum degree of a vertex in $G$. Finally, we present two kernels of sizes $O(k \cdot m(f))$ and $O(k \cdot Δ(G))$.

Figures (6)

• Figure 1: $G'$ is the graph resulting from the construction applied to the graph $G$.
• Figure 2: The vertices in $v_1$ and $v_3$ are positioned in the circumference of $v$ precisely where the incident edges touch $v$, assuming $d_G(v) = 2$ and $\Delta(G)=3$. The other vertices are positioned between or after them. In black, we have the subgraph $G'[P_v \cup Q_v]$.
• Figure 3: The overall structure of $G'$ resulting from the construction applied to the path $P = v_1,v_2,v_3$ and $k=3$, with $Z \cup W$ omitted. Unlike gray edges, a black edge between an individual vertex $x$ (white colored nodes) and a set of vertices $Q$ (gray colored nodes) does not necessarily represent an edge between $x$ and every vertex in $Q$.
• Figure 4: Part of \ref{['fig:reductiontreewidth']} in more details. It depicts the vertices in $U^1$, $U^2$, $C^{1,2}$, $C^{2,1}$ and $Y^{1,2}$, the vertices $u^1$, $u^2$ and $y^{1,2}$ and the edges between them. For clarity, some edges are in light gray.
• Figure 5: Gadget added to $G'$$m(f) - f(v)$ times for each $v \in V$ such that $f(v) < m(f)$.

Theorems & Definitions (31)

• Theorem 1
• proof
• Claim 2
• Theorem 3
• proof
• Claim 4
• Claim 5
• Claim 6
• Theorem 7
• proof