Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Perfect Roman Domination and Unique Response Roman Domination

The idea of enumeration algorithms with polynomial delay is to polynomially bound the running time between any two subsequent solutions output by the enumeration algorithm. While it is open for more than four decades if all minimal dominating sets of a graph can be enumerated in output-polynomial time, it has recently been proven that pointwise-minimal Roman dominating functions can be enumerated even with polynomial delay. The idea of the enumeration algorithm was to use polynomial-time solvable extension problems. We use this as a motivation to prove that also two variants of Roman dominating functions studied in the literature, named perfect and unique response, can be enumerated with polynomial delay. This is interesting since Extension Perfect Roman Domination is W[1]-complete if parameterized by the weight of the given function and even W[2]-complete if parameterized by the number vertices assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability of extension problems and enumerability with polynomial delay tend to go hand-in-hand. We achieve our enumeration result by constructing a bijection to Roman dominating functions, where the corresponding extension problem is polynomimaltime solvable. Furthermore, we show that Unique Response Roman Domination is solvable in polynomial time on split graphs, while Perfect Roman Domination is NP-complete on this graph class, which proves that both variations, albeit coming with a very similar definition, do differ in some complexity aspects. This way, we also solve an open problem from the literature

Signed double Roman domination on cubic graphs

The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled $\pm{}1$ have at least one neighbor with label in $\{2,3\}$; (ii) each vertex labeled $-1$ has one $3$-labeled neighbor or at least two $2$-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order $n$ for which we present a sharp $n/2+\Theta(1)$ lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic $2\times m$ grid graphs, among other results

Chiraptophobic cockroaches evading a torch light

We propose and study game-theoretic versions of independence in graphs. The games are played by two players - the aggressor and the defender - taking alternate moves on a graph G with tokens located on vertices from an independent set of G. A move of the aggressor is to select a vertex v of G. A move of the defender is to move tokens located on vertices in NG(v) each along one incident edge. The goal of the defender is to maintain the set of occupied vertices independent while the goal of the aggressor is to make this impossible. We consider the maximum number of tokens for which the aggressor can not win in a strategic and an adaptive version of the game

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum