On αrγs(k)-perfect graphs

AbstractFor some integer k⩾0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)⩽k for every induced subgraph H of G. For r⩾1 let αr and γr denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α1γ1(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study αrγs(k)-perfect graphs for r,s⩾1. We prove several properties of minimal αrγs(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α2r−1γr(k)-perfect graphs for r⩾1. Furthermore, we characterize claw-free α2γ2(k)-perfect graphs

Total domination versus paired domination

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by gamma_t . The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Gamma_t . A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by gamma_p . The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Gamma_p . In this paper we prove several results on the ratio of these four parameters: For each r ge 2 we prove the sharp bound gamma_p/gamma_t le 2 - 2/r for K_{1,r} -free graphs. As a consequence, we obtain the sharp bound gamma_p/gamma_t le 2 - 2/(Delta+1) , where Delta is the maximum degree. We also show for each r ge 2 that {C_5,T_r} -free graphs fulfill the sharp bound gamma_p/gamma_t le 2 - 2/r , where T_r is obtained from K_{1,r} by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Gamma_p / Gamma_t . Further, we prove that a graph hereditarily has an induced paired dominating set iff gamma_p le Gamma_t holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio gamma_p / Gamma_t taken over the induced subgraphs of a graph. As a consequence, we prove for each r ge 3 the sharp bound gamma_p/Gamma_t le 2 - 2/r for graphs that do not contain the corona of K_{1,r} as subgraph. In particular, we obtain the sharp bound gamma_p/Gamma_t le 2 - 2/Delta

Total domination versus paired domination

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by gamma_t . The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Gamma_t . A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by gamma_p . The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Gamma_p . In this paper we prove several results on the ratio of these four parameters: For each r ge 2 we prove the sharp bound gamma_p/gamma_t le 2 - 2/r for K_{1,r} -free graphs. As a consequence, we obtain the sharp bound gamma_p/gamma_t le 2 - 2/(Delta+1) , where Delta is the maximum degree. We also show for each r ge 2 that {C_5,T_r} -free graphs fulfill the sharp bound gamma_p/gamma_t le 2 - 2/r , where T_r is obtained from K_{1,r} by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Gamma_p / Gamma_t . Further, we prove that a graph hereditarily has an induced paired dominating set iff gamma_p le Gamma_t holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio gamma_p / Gamma_t taken over the induced subgraphs of a graph. As a consequence, we prove for each r ge 3 the sharp bound gamma_p/Gamma_t le 2 - 2/r for graphs that do not contain the corona of K_{1,r} as subgraph. In particular, we obtain the sharp bound gamma_p/Gamma_t le 2 - 2/Delta

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

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