We give a maximal independent set (MIS) algorithm that runs in $O(\log \log \Delta)$ rounds in the congested clique model, where $\Delta$ is the maximum degree of the input graph. This improves upon the $O(\frac{\log(\Delta) \cdot \log \log \Delta}{\sqrt{\log n}} + \log \log \Delta )$ rounds algorithm of [Ghaffari, PODC '17], where $n$ is the number of vertices of the input graph. In the first ...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V \ S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in S. The total domination number of a graph...

A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. We show that given a graph of order n with minimum degree at least 2, one can add at most (n−2√n )/4+O(log n) edges such that the resulting graph has two disjoint total dominating sets, and this bound is best possible.

We present a unified approach to Doob’s Lp maximal inequalities for 1 ≤ p < ∞. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob’s maximal inequalities. These are best possib...

We give an 0 (nlogln) [ime algorirnm for finding a minimum independem dominating se[ in a pennmation graph. TItis improves on ilie previous D(n 3) time algorictun known for solving tllis problem [4]. .,. Dept of CompUlcr Sci., Purdue Univ., West Gf:tyelle, IN 47907. Rcso::trch ~upported by ONR. Contr:lct NOOOI-l-34-K. 0502:md NSF Gl':tnl DCR-8451393, wilh matching funds from AT&T. • o.:pt of Ma...

A graph on n vertices which is the union of two comparability graphs on the same vertex set, always contains a clique or independent set of size n 1 3 . On the other hand, there exist such graphs for which the largest clique and independent set are of size at most n. Similar results are obtained for graphs which are a union of a fixed number k of comparability graphs. We also show that the same...

For a graph G = (V,E), a set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S as well as another vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. In this paper we find all graphs G satisfying γr(G) = n− 3, where n is the order of G.

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number t(G) is the minimum cardinality of a total dominating set of G. Let G be a connected spanning subgraph of Ks;s, and let H be the complement of G relative to Ks;s; that is, Ks;s = G ⊕ H is a factorization of Ks;s. The graph G is k-supercritical relative...

Let F be a family of translates of a fixed convex set M in Rn. Let (F) and (F) denote the transversal number and the independence number of F, respectively. We show that (F) (F) 8 (F) − 5 for n = 2 and (F) 2n−1nn (F) for n 3. Furthermore, if M is centrally symmetric convex body in the plane, then (F) (F) 6 (F)− 3. © 2006 Elsevier B.V. All rights reserved.