Cops and Robbers playing on edges

Dudek, Andrzej; Gordinowicz, Przemysław; Prałat, Paweł

doi:10.4310/joc.2014.v5.n1.a6

Cited by 11 publications

(17 citation statements)

References 22 publications

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“…Proof (Theorem 1.3). As mentioned in the introduction, the class LINE of line graphs has unbounded cop number by results of Dudek et al [7]. Moreover, each line graph can be represented as the intersection graph of two-element subsets of the real line, and therefore the classes 2-UNIT INTERVAL and 2-INTERVAL contain LINE as a subclass.…”

Section: Unbounded Cop Number Of Intersection Graphs Of Disconnected mentioning

confidence: 97%

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Cops and Robbers on intersection graphs

Gavenčiak

Gordinowicz

Jelínek

et al. 2018

European Journal of Combinatorics

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The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs:• The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g . For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

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Section: Unbounded Cop Number Of Intersection Graphs Of Disconnected mentioning

confidence: 97%

“…Thus, line-graphs provide a simple example of intersection graphs of disconnected sets. As shown by Dudek et al [7], the cop number of L(G) is related to the cop number of G via the inequalities cn(G) 2 ≤ cn(L(G)) ≤ cn(G) + 1.…”

mentioning

confidence: 97%

Cops and Robbers on intersection graphs

Gavenčiak

Gordinowicz

Jelínek

et al. 2018

European Journal of Combinatorics

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show abstract

“…As mentioned in the introduction, the class LINE of line graphs has unbounded cop number by results of Dudek et al [7]. Moreover, each line graph can be represented as the intersection graph of two-element subsets of the real line, and therefore the classes 2-UNIT INTERVAL and 2-INTERVAL contain LINE as a subclass.…”

Section: Unbounded Cop Number Of Intersection Graphs Of Disconnected mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

Cops and Robbers on Intersection Graphs

Gavenčiak

Jelínek

Klavík

et al. 2013

Algorithms and Computation

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The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs:• The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g . For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4.• The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

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“…There also exist cases (for example moving in a maze or a road network) where it makes more sense to assume that both cops and robber move from edge to edge. We will call the classical version of the edge CR game edge av-CR; it has attracted attention only recently [28]. Edge ai-CR, dv-CR and di-CR variants are also possible, in analogy to the node versions listed in the Table. Each of these cases can be reduced to the corresponding node variant, with the edge game taking place on the line graph L (G) of G.…”

Section: The Cr Game Familymentioning

confidence: 99%

The Role of Visibility in Pursuit/Evasion Games

2014

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Abstract:The cops-and-robber (CR) game has been used in mobile robotics as a discretized model (played on a graph G) of pursuit/evasion problems. The "classic" CR version is a perfect information game: the cops' (pursuer's) location is always known to the robber (evader) and vice versa. Many variants of the classic game can be defined: the robber can be invisible and also the robber can be either adversarial (tries to avoid capture) or drunk (performs a random walk). Furthermore, the cops and robber can reside in either nodes or edges of G. Several of these variants are relevant as models or robotic pursuit/evasion. In this paper, we first define carefully several of the variants mentioned above and related quantities such as the cop number and the capture time. Then we introduce and study the cost of visibility (COV), a quantitative measure of the increase in difficulty (from the cops' point of view) when the robber is invisible. In addition to our theoretical results, we present algorithms which can be used to compute capture times and COV of graphs which are analytically intractable. Finally, we present the results of applying these algorithms to the numerical computation of COV.

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Cops and Robbers playing on edges

Cited by 11 publications

References 22 publications

Cops and Robbers on intersection graphs

Cops and Robbers on intersection graphs

Cops and Robbers on Intersection Graphs

The Role of Visibility in Pursuit/Evasion Games

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