The one-visibility Localization game

TL;DR

A variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the one-visibility localization number $\zeta_1$ is introduced, showing that the parameter is unbounded on k-ary trees.

Abstract

We introduce a variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the one-visibility localization number $ζ_1$. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for $ζ_1$ on $k$-ary trees with $k\ge 2$ that differ by a multiplicative constant, showing that the parameter is unbounded on $k$-ary trees. We provide a $O(\sqrt{n})$ bound for $K_h$-minor free graphs of order $n$, and we show Cartesian grids meet this bound by determining their one-visibility localization number up to four values. We present upper bounds on $ζ_1$ using pathwidth and the domination number and give upper bounds on trees via their depth and order. We conclude with open problems.

Figures (4)

• Figure 1: A graph of $\Phi(G,k)$ that illustrates $H_V(G)$, where a contiguous set of $H_V(G)$ integers each have $\Phi(G,k) \geq H_V(G)$. For Theorem \ref{['thm:h_index']}, if the cops manage to reduce the number of contaminated vertices $k$ below $k_b+1$ (in our example, moving from $k=k_1$ to $k=k_2$ that is in the gray region), then the contamination is guaranteed to grow to some cardinality at least $k_b+1$ (that is, moving from $k=k_2$ to $k=k_3$ in our example, which is to the right of the gray region).
• Figure 2: Possible robber locations are shown in red after $C_1$'s third probe.
• Figure 3: Cops placed on every other column can clear two rows.
• Figure 4: The first sweep of claim 1 where $m=7$. Red and pink dots indicate the forced region before the cops' move, and the square indicates the $i,j$ such that this forced region is $F_{i,j}$. Red dots indicate the forced region after the cops' move. Black circles indicate the locations of the cops their move. From left to right, top to bottom, the images indicate play just before and after the cops' 1st, 4th, 6th, 9th, 11th, 14th, 16th, and 19th move.

Theorems & Definitions (50)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Lemma 5
• Theorem 6