We introduce a new variant of the Goemans-Williamson (GW) scheme for the Prize-Collecting Steiner Tree Problem (PCST). Motivated by applications in signal processing, the focus of our contribution is to construct a very fast algorithm for the PCST problem that still achieves a provable approximation guarantee. Our overall algorithm runs in time O(dm logn) on a graph with m edges, where all edge...

The domination polynomial D(G, x) of a graph G is the generating function of its dominating sets. We prove that D(G, x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G, x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G, x) based on articulation vertices, and more generally, on splitting sets of vertices.

Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P , and EPT(P) is the EPT graph (i.e. the edge intersection graph) of P . These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their u...

Incremental closure is a proof search technique for free variable analytic tableaux where partial solutions are updated along with tableaux expansion. In this paper we address possible adaptations of the incremental closure technique to variable splitting tableaux. Variable splitting tableaux provide a sound way to instantiate occurrences of the same variable in different branches differently b...

We give counterexamples to two conjectures in \Bill Jackson, Some remarks on arc-connectivity, vertex splitting, and orientation in graphs and digraphs, Journal of Graph Theory, 12(3):429{436, 1988" concerning orientations of mixed graphs and splitting oo in digraphs, and prove the rst conjecture in the (di-) Eulerian case(s). Beside that we solve the degree constrained directed augmentation pr...

We show that, in any undirected graph, splitting off can be performed while preserving all cuts of value at most 4/3 times the minimum value, and this is best possible. This generalizes a classical splitting-off result of Lovász. CORE, University of Louvain, Belgium. This work was partially supported by DONET (European Community contract number ERB FMRX-CT98-0202) and NSF (contract 9623859-CCR).

A spanning tree of graph G is a spanning subgraph of G that is a tree. In this paper, we focus our attention on (n,m) graphs, where m = n, n + 1, n + 2, n+3 and n + 4. We also determine some coefficients of the Laplacian characteristic polynomial of fullerene graphs.

let $g$ be a simple graph with vertex set $v(g) = {v_1, v_2,ldots, v_n}$ and edge set $e(g) = {e_1, e_2,ldots , e_m}$. similar tothe randi'c matrix, here we introduce the randi'c incidence matrixof a graph $g$, denoted by $i_r(g)$, which is defined as the$ntimes m$ matrix whose $(i, j)$-entry is $(d_i)^{-frac{1}{2}}$ if$v_i$ is incident to $e_j$ and $0$ otherwise. naturally, therandi'c incidenc...