Related papers: $k$-Hyperopic Cops and Robber

We present two efficient algorithms that compute the optimal strategy for cop in the game of Cop v.s. Gambler where the gambler's strategy is not optimal but known to the cop. The first algorithm is analogous to Bellman-Ford algorithm for…

We prove that every connected $P_5$-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected $P_5$-free graph $G$ with independence number at least three contains a…

In the Hunters and Rabbit game, $k$ hunters attempt to shoot an invisible rabbit on a given graph $G$. In each round, the hunters select $k$ vertices to shoot at, while the rabbit moves along an edge of $G$. The hunters win if, at any…

We present an algorithm of time complexity $O(kn^{k+2})$ deciding whether a graph $G$ on $n$ vertices is $k$-copwin. The fastest algorithm thus far had time complexity $O(n^{2k+2})$.

We study the localization number of incidence graphs of designs. In the localization game played on a graph, the cops attempt to determine the location of an invisible robber via distance probes. The localization number of a graph $G$,…

In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph $\mathcal{H}$ by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the…

We introduce and study the Generalized Cops and Robbers game (GCR), an N-player pursuit game in graphs. The two-player version is essentially equivalent to the classic Cops and Robbers (CR) game. The three-player version can be understood…

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move,…

The domatic number of a graph is the maximum number of pairwise disjoint dominating sets admitted by the graph. We introduce a game based around this graph invariant. The domatic number game is played on a graph $G$ by two players, Alice…

This paper considers the Hamiltonian walk problem in the multi-agent coordination framework, referred to as $k$-agents Hamiltonian walk problem ($k$-HWP). In this problem, a set of $k$ connected agents collectively compute a spanning walk…

We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio~(2021) proved that two…

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are…

We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel's conjecture states that for every connected graph $G$ on $n$ vertices, the cop number of $G$ is upper bounded by $O(\sqrt{n})$,…

We introduce and study the game of "Selfish Cops and Active Robber" (SCAR) which can be seen as an multiplayer variant of the "classic" two-player Cops and Robbers (CR) game. In classic CR all cops are controlled by a single player, who has…

In previous work, we introduced median decompositions, a generalisation of tree decompositions where a graph can be modelled after any median graph, along with a hierarchy of $i$-medianwidth parameters $(mw_i)_{i\geq 1}$ starting from…

In the $\left(1:b\right)$ component game played on a graph $G$, two players, Maker and Breaker, alternately claim~$1$ and~$b$ previously unclaimed edges of $G$, respectively. Maker's aim is to maximise the size of a largest connected…

In a Maker-Breaker game on a graph $G$, Breaker and Maker alternately claim edges of $G$. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker-Breaker games…

The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing…

In the Localization game played on graphs, a set of cops uses distance probes to identify the location of an invisible robber. We present an extension of the game and its main parameter, the localization number, to directed graphs. We…

We study the m-Eternal Domination problem, which is the following two-player game between a defender and an attacker on a graph: initially, the defender positions k guards on vertices of the graph; the game then proceeds in turns between…