A block-chain is a graph whose block graph is a path, i.e. it is either a P1, a P2, or a 2-connected graph, or a graph of connectivity 1 with exactly two end-blocks. A graph is called traceable if it contains a Hamilton path. A traceable graph is clearly a block-chain, but the reverse does not hold in general. In this paper we characterize all pairs of connected graphs {R, S} such that every {R...

Given a graphG, the k-dominating graph ofG, Dk(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in Dk(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph Dk(G) aids in studying the reconfiguration problem for dominating sets. In parti...

Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoičić (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the Szemerédi-Trotter Theorem ...

A graph G is a chordal-k-generalized split graph if G is chordal and there is a clique Q in G such that every connected component in G[V \ Q] has at most k vertices. Thus, chordal-1-generalized split graphs are exactly the split graphs. We characterize chordal-k-generalized split graphs by forbidden induced subgraphs. Moreover, we characterize a very special case of chordal-2-generalized split ...

Let k¿2; n=2+1; and let m0; : : : ; mk−1 each be a multiple of n. The graph Cm0×· · ·×Cmk−1 consists of isomorphic connected components, each of which is (n − 1)-regular and admits of a vertex partition into n smallest independent dominating sets. Accordingly, (independent) domination number of each connected component of this graph is equal to (1=n)th of the number of vertices in it. ? 2001 El...

Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M , we conjecture that if the Tate cohomology Ĥ ∗ (G, M) of G with coefficients in M is finitely generated over the Tate cohomology ring Ĥ ∗ (G, k), then the support variety VG(M) of M is equal to the entire maximal ideal spectrum VG(k). We prove various results w...

In the sequel, we suppose that p(l) = 2, p(2), p(3),... is the sequence of primes in ascending order. Let T be a rooted tree, r its root. The connected components of Tr are denoted by T, ,..., T,, where v is the degree of r. The graphs Tj ul,..., v) obviously are trees, which we transform into rooted trees by defining as the root of Tj the vertex of 7” which is adjacent to r in T. We now assign...

This paper gives a survey of some recent results, most of them available in print and the rest still unpublished, in confidence bounds on parametric f'unctions associated with multivariate normal distributions, that have been developed with a particular end in view. The general philosophy is discussed in greater detail than heretofore, and most of the results obtained so far are exhibited in th...

Let f : CP 99K CP be a rational map with algebraic and topological degrees both equal to d ≥ 2. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms T : CP → CP, the perturbed map T ◦f admits exactly two ergodic measures of maximal entropy log d, one of saddle and one of repelling type. Neither measure is supported in...

A set D of vertices in a graph G = (V, E) is a dominating set of G if every vertex in V −D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. We define the cobondage number bc(G) of G to be the minimum cardinality among the sets of edges X ⊆ P2(V ) −E, where P2(V ) = {X ⊆ V : |X| = 2} such that γ(G + X) < γ(G). In this paper, the exa...