On the Cop Number of String Graphs

TL;DR

The paper advances the Cops and Robber game on string graphs by establishing a tight bound of $\mathsf{c}(G)\le 13$ using a novel guarding technique that leverages convex isometric paths and geometric robber territories. It also shows $\mathsf{c_a}(G)\le 4$ for planar graphs, and derives corollaries for $2$-BOX graphs and the chromatic number of girth-$5$ string graphs, while providing an algorithmic route to decide cop numbers in $O(n^{15})$ time. The methods rely on a detailed correspondence between string representations and guarded regions, extending isometric/convex-path arguments to a global strategy that iteratively reduces the robber’s region until capture. These results not only improve bounds for string and related graph classes but also offer techniques applicable to Fully Active variants and other representation-based pursuit games. The work highlights deep connections between geometric representations, guard strategies, and combinatorial bounds, with potential impact on chromatic and width-parameter analyses in intersection-graph settings.

Abstract

Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The \emph{cop number} of a graph is the minimum number of cops required to capture the robber. Gavenčiak et al.~[Eur. J. of Comb. 72, 45--69 (2018)] studied the game on intersection graphs and established that the cop number for the class of string graphs is at most 15, and asked as an open question to improve this bound for string graphs and subclasses of string graphs. We address this question and establish that the cop number of a string graph is at most 13. To this end, we develop a novel \textit{guarding} technique. We further establish that this technique can be useful for other Cops and Robber games on graphs admitting a representation. In particular, we show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al.~[Austr. J. Comb. 76(2), 248--265 (2020)]. In passing, we also improve the known bounds on the cop number of boxicity 2 graphs. Finally, as a corollary of our result on the cop number of string graphs, we establish that the chromatic number of string graphs with girth at least $5$ is at most $14$.

Figures (7)

• Figure 1: The dodecahedron and its boxicity 2 representation. Here each vertex $i$ corresponds to rectangle $i$.
• Figure 2: $P$ is a convex $u_0,u_k$ path (relative to $T$). Here $N[P]$ is denoted by heavier bold lines. The vertex $x\in (N[P]\cap D_i)\setminus\{u_i\}$. The only two possible edges between $x$ and vertices of $P$ are denoted by dotted lines.
• Figure 3: Here (a) represents $\Psi$ and (b) represents $\Psi_B$. $\psi(v)$ is not in $\Psi_B$, $\psi(z)$ is in $\Psi_B$. Further, for $\psi(x)$, strings $\psi(x_1)$, $\psi(x_2)$, and $\psi(x_3)$ are in $\Psi_B$; for $\psi(y)$, strings $\psi(y_1)$ and $\psi(y_2)$ are in $\Psi_B$; and for $\psi(w)$, the string $\psi(w_1)$, which is a single point, is in $\Psi_B$.
• Figure 4: Illustration of geometric robber territory. Here the boundary of $B$ is depicted in bold black, the strings in $\Psi_B$ that do not intersect with boundary of $B$ are depicted in green, and the strings that intersect with the boundary of $B$ are depicted in red. $\Psi_B$ is the geometric robber territory if: (i) $\mathcal{R}$ is on a green string, and (ii) $\mathcal{R}$ gets captured if it moves to a red string. Although parts of red strings are in $\Psi_B$, none of these strings are accessible to $\mathcal{R}$.
• Figure 5: In both (a) and (b), the curves contain points to highlight the segments that form them. In both subfigures, $\mathcal{R}$ is on a green string and cannot access a red string without getting captured immediately.

Theorems & Definitions (33)

• Proposition 1: aigner
• Proposition 2: gavenciak
• Lemma 2
• Theorem 3
• Corollary 3
• Proposition 4: gavenciak
• Corollary 4
• Theorem 5
• proof
• Lemma 5