Defensive alliances in graphs: a survey

A set $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-alliance free sets.Comment: 25 page

The Price of Connectivity for Vertex Cover

The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the {\em Price of Connectivity}, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number $t$. We obtain forbidden induced subgraph characterizations for every real value $t \leq 3/2$. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most $t$. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class $\Theta_2^P = P^{NP[\log]}$, the class of decision problems that can be solved in polynomial time, provided we can make $O(\log n)$ queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.Comment: 19 pages, 8 figure

Random interlacements and amenability

We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039-2087] for the special case of ${\mathbb{Z}}^d$ (with $d\geq3$). In Sznitman [Ann. of Math. (2) (2010) 171 2039-2087], it was shown that on ${\mathbb{Z}}^d$: for any intensity $u>0$, the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity u the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of u. Finally, we establish the monotonicity of the transition between the "disconnected" and the "connected" phases, providing the uniqueness of the critical value $u_c$ where this transition occurs.Comment: Published in at http://dx.doi.org/10.1214/12-AAP860 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org