McCuaig and Ota proved that every 3-connected graph G on at least 9 vertices admits a contractible triple, i. e. a connected subgraph H on three vertices such that G − V (H) is 2-connected. Here we show that every 3-connected graph G on at least 9 vertices has more than |V (G)|/10 many contractible triples. If, moreover, G is cubic, then there are at least |V (G)|/3 many contractible triples, w...

Definition 1. Given a undirected graph G = (V,E), two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = (v1, v2, . . . , vn) ∈ V n such that vi is adjacent to vi+1 for 1 ≤ i < n. Such a path P is called a path of length n from v1 to vn. Let ei,j be the edge incident to both vi and vj. Given a real-valued weight functi...

Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that...

Given a graph G = (V,E) with a positive weight function w on the vertices of G, a global powerful alliance of G is a subset S of V such that for every vertex v at least half of the total weight in the closed neighborhood of v is contributed by the vertices of S. Finding the smallest such set in general graphs is NP-complete, even when the weights are all the same. In this paper, we give a linea...

We prove that there is no cubic graph with diameter 4 on 40 vertices. This implies that the maximal number of vertices of a (3,4)-graph is 38. ? 2000 Elsevier Science B.V. All rights reserved.

Let  denote the positive octant of the regular -dimensional cubic lattice. Each vertex (1 2     ) of  is adjacent to all vertices of the form (1 2      + 1     ), 1 ≤  ≤ . A -partition of a positive integer  is an assignment of nonnegative integers 12 to the vertices of , subject to both an ordering condition 12 ≥ max 1≤≤ 12+...

A vertex|matching-partition (V |M) of a simple graph G is a spanning collection of vertices and independent edges of G. Let vertex v ∈ V have weight wv and edge e ∈ M have weight we. Then the weight of V |M is w(V |M) = ∏ v∈V wv · ∏ e∈M we. Define the vertex|matching-partition function of G as W(G) = ∑ V |M w(V |M). In this paper we study this function when G is a path and a cycle. We generate ...

We estimate convex polytopes and general convex sets in R, d ≥ 2 in the regression framework. We measure the risk of our estimators using a L-type loss function and prove upper bounds on these risks. We show that, in the case of polytopes, these estimators achieve the minimax rate. For polytopes, this minimax rate is lnn n , which differs from the parametric rate for non-regular families by a l...

A dominating set in a graph G is a set S of vertices such that every vertex outside S has a neighbor in S; the domination number γ(G) is the minimum size of such a set. The independent domination number, written i(G), is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured that if G is a connected cubic graph with sufficiently many vertices, then i(G)...

An antimedian of a profile π = (x1, x2, . . . , xk) of vertices of a graph G is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on G and has as output the set of antimedians of a profile. It is a typical location function for finding a location for an obnoxious facility. The ‘converse’ of the antimedian f...