A Connected Version of the Graph Coloring Game

The graph coloring game is a two-player game in which, given a graph G and a set of k colors, the two players, Alice and Bob, take turns coloring properly an uncolored vertex of G, Alice having the first move. Alice wins the game if and only if all the vertices of G are eventually colored. The game chromatic number of a graph G is then defined as the smallest integer k for which Alice has a winning strategy when playing the graph coloring game on G with k colors. In this paper, we introduce and study a new version of the graph coloring game by requiring that, after each player's turn, the subgraph induced by the set of colored vertices is connected. The connected game chromatic number of a graph G is then the smallest integer k for which Alice has a winning strategy when playing the connected graph coloring game on G with k colors. We prove that the connected game chromatic number of every outerplanar graph is at most 5 and that there exist outerplanar graphs with connected game chromatic number 4. Moreover, we prove that for every integer k ≥ 3, there exist bipartite graphs on which Bob wins the connected coloring game with k colors, while Alice wins the connected coloring game with two colors on every bipartite graph

Scalable Approximation Algorithm for Network Immunization

The problem of identifying important players in a given network is of pivotal importance for viral marketing, public health management, network security and various other fields of social network analysis. In this work we find the most important vertices in a graph G = (V;E) to immunize so as the chances of an epidemic outbreak is minimized. This problem is directly relevant to minimizing the impact of a contagion spread (e.g. flu virus, computer virus and rumor) in a graph (e.g. social network, computer network) with a limited budget (e.g. the number of available vaccines, antivirus software, filters). It is well known that this problem is computationally intractable (it is NP-hard). In this work we reformulate the problem as a budgeted combinational optimization problem and use techniques from spectral graph theory to design an efficient greedy algorithm to find a subset of vertices to be immunized. We show that our algorithm takes less time compared to the state of the art algorithm. Thus our algorithm is scalable to networks of much larger sizes than best known solutions proposed earlier. We also give analytical bounds on the quality of our algorithm. Furthermore, we evaluate the efficacy of our algorithm on a number of real world networks and demonstrate that the empirical performance of algorithm supplements the theoretical bounds we present, both in terms of approximation guarantees and computational efficiency

From Pathwidth to Connected Pathwidth

It is proven that the connected pathwidth of any graph $G$ is at most 2\cdot\pw(G)+1, where \pw(G) is the pathwidth of $G$. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width $k$ computes a connected path decomposition of width at most $2k+1$. The running time of the algorithm is $O(dk^2)$, where $d$ is the number of `bags' in the input path decomposition. The motivation for studying connected path decompositions comes from the connection between the pathwidth and the search number of a graph. One of the advantages of the above bound for connected pathwidth is an inequality \csn(G)\leq 2\sn(G)+3, where \csn(G) and \sn(G) are the connected search number and the search number of $G$. Moreover, the algorithm presented in this work can be used to convert a given search strategy using $k$ searchers into a (monotone) connected one using $2k+3$ searchers and starting at an arbitrary homebase

Contraction Obstructions for Connected Graph Searching

We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the question whether the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for $k=2$, where $k$ is the number of searchers we are allowed to use. This set is finite, it consists of 177 graphs and completely characterises the graphs with connected (and monotone) mixed search number at most 2. Our proof reveals that the "sense of direction" of an optimal search searching is important for connected search which is in contrast to the unconnected original case. We also give a double exponential lower bound on the size of the obstruction set for the classes where this set is finite