Independent domination in hereditary classes

AbstractWe investigate Independent Domination Problem within hereditary classes of graphs. Boliac and Lozin [Independent domination in finitely defined classes of graphs, Theoret. Comput. Sci. 301 (1–3) (2003) 271–284] proved some sufficient conditions for Independent Domination Problem to be NP-complete within finitely defined hereditary classes of graphs. They posed a question whether the conditions are also necessary. We show that the conditions are not necessary, since Independent Domination Problem is NP-hard within 2P3-free graphs.Moreover, we show that the problem remains NP-hard for a new hereditary class of graphs, called hereditary 3-satgraphs. We characterize hereditary 3-satgraphs in terms of forbidden induced subgraph. As corollaries, we prove that Independent Domination Problem is NP-hard within the class of all 2P3-free perfect graphs and for K1,5-free weakly chordal graphs.Finally, we compare complexity of Independent Domination Problem with that of Independent Set Problem for a hierarchy of hereditary classes recently proposed by Hammer and Zverovich [Construction of maximal stable sets with k-extensions, Combin. Probab. Comput. 13 (2004) 1–8]. For each class in the hierarchy, a maximum independent set can be found in polynomial time, and the hierarchy covers all graphs. However, our characterization of hereditary 3-satgraphs implies that Independent Domination Problem is NP-hard for almost all classes in the hierarchy. This fact supports a conjecture that Independent Domination is harder than Independent Set Problem within hereditary classes

Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with 2g vertices. The mapping class group over the graph Gamma is defined to be a subgroup of the automorphism group of the right-angled Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of Magnus.Comment: 45 page

Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

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