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Guarding a subgraph as a tool in pursuit-evasion games
Drago Bokal, Janja Jerebic, 2021, original scientific article

Abstract: Pursuit-evasion games study the number of cops needed to capture therobber in a game played on a graph, in which the cops and the robber movealternatively to neighbouring vertices, and the robber is captured if a copsteps on the vertex the robber is in. A common tool in analyzing this copnumber of a graph is a cop moving along a shortest path in a graph, thuspreventing the robber to step onto this path. We generalize this approach byintroducing a shadow of the robber, the maximal set of vertices from whichthe cop parries the protected subgraph. In this context, the robber becomesan intruder and the cop becomes the guard. We show that the shadow canbe computed in polynomial time, implying polynomial time algorithms forcomputing both a successful guard as well as a successful intruder, whicheverexists. Furthermore, we show that shadow function generalizes the conceptof graph retractions. In some cases, this implies a polynomially computablecertification of the negative answer to the NP-complete problem of existenceof a retraction to a given subgraph.
Keywords: pursuit-evasion game, graph searching, guarding, shadow function, graph retraction
Published in DKUM: 17.08.2023; Views: 314; Downloads: 25
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On cubic and edge-critical isometric subgraphs of hypercubes
C. Paul Bonnington, Sandi Klavžar, Alenka Lipovec, 2002

Abstract: Predstavljene so vse kubične delne kocke do 30 točk in vse po povezavah kritične delne kocke do 14 točk. Seznama sta bila potrjena z računalniškim iskanjem. Konstruirane so tudi netrivialne kubične delne kocke na 36, 42 in 48 točkah.
Keywords: matematika, teorija grafov, delna kocka, hiperkocka, kubični graf, računalniško iskanje, mathematics, graph theory, partial cube, hypercube, cubic graph, computer searching
Published in DKUM: 10.07.2015; Views: 1113; Downloads: 32
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