An approximation algorithm for zero forcing

Abstract

We give an algorithm that finds a zero forcing set which approximates the optimal size by a factor of $\text{pw}(G)+1$, where $\text{pw}(G)$ is the pathwidth of $G$. Starting from a path decomposition, the algorithm runs in $O(nm)$ time, where $n$ and $m$ are the order and size of the graph, respectively. As a corollary, we obtain a new upper bound on the zero forcing number in terms of the fort number and the pathwidth. The algorithm is based on a correspondence between zero forcing sets and forcing arc sets. This correspondence leads to a new bound on the zero forcing number in terms of vertex cuts, and to new, short proofs for known bounds on the zero forcing number.

Figures (2)

• Figure 1: A set of dipaths that contains a chain twist
• Figure 2: An illustration for the proof of the converse direction of Theorem \ref{['thm:chaintwist']}. The figure consists of $H = (V, A)$ that satisfies \ref{['cond:fas1']}, and the remaining edges in $G$ are omitted. Filled vertices correspond to blue vertices and unfilled vertices correspond to white vertices. Each dipath of $H$ that is not all blue consists of an initial segment of blue vertices, indicated by a squiggly arrow, followed by an arc $(b_i,v_i)$, indicated by a thin arrow, followed by a final segment of white vertices, indicated by a dashed arrow. Thin lines represent edges that are not in $A$. In this case, we form a digraph $D$ with a vertex set $\{1,2,3,4\}$ and an edge set $\{(1,2), (2,3), (3,4), (4,2)\}$ where there is one directed cycle on $2,3,4$.

Theorems & Definitions (47)

• Definition 2.1
• Definition 2.2: Chain twist
• Remark 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Proposition 3.1
• proof