Mobile Eternal Security in Graphs

We consider the problem of securing a graph against a sequence of vertex attacks by placing guards on its vertices. The guards are “mobile” in that, after any attack, each guard can move to any neighbor of its current location, but one guard must move to the attacked vertex. We determine sharp upper bounds, in terms of the order of the graph, on the minimum number of guards necessary for connected graphs and graphs with minimum degree at least 2. We also derive sharp Nordhaus-Gaddum type bounds. There are many different notions for the “security” of a structure, which depend heavily on how the structure is attacked and how it can be protected. When the structure in question is a graph, several attack/protection schemes have already been studied [2, 3, 4, 5, 6]. In this paper, we study a newer protection scheme introduced in [7]. In this model, mobile guards are placed on the vertices of a graph. The guards can move to any neighbor of their current location in any time step. Research partially supported by an NSF Graduate Research Fellowship and by NSF grant DMS-0528086. Department of Computer Science, University of Illinois, 201 N. Goodwin Ave., Urbana, IL 61801-2302. Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801.