Given a weighted planar graph, we construct a simple separating cycle C whose interior and exterior both weigh at most 2/3 of the total weight. The cycle C uses k jumps across the faces and ordinary edges with cost O(1=k) of the total edge cost. This theorem is the main component of a PTAS for the weighted TSP problem 4].

Let G be a simple, undirected graph and Gw be the positive weighted graph obtained from G by giving weights to its edges using the positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. Let G denote the class of connected, unweighted, bipartite graphs G with a unique perfect matching M such that G/M (the graph obtained by contract...

In this article, only connected bipartite graphs G with a unique perfect matching M are considered. Let Gw denote the weighted graph obtained from G by giving weights to its edges using the positive weight function w : E(G)→ (0,∞) such that w(e) = 1 for each e ∈ M. An unweighted graph G may be viewed as a weighted graph with the weight function w ≡ 1 (all ones vector). A weighted graph Gw is no...

Weighted graph matching is computationally challenging due to the combinatorial nature of the set of permutations. In this paper, a new relaxation approach to weighted graph matching is proposed, by which a new matching algorithm, named alternate iteration algorithm, is designed. It is proved that the algorithm proposed is locally convergent. Experiments are presented to show the effectiveness ...

We present a decomposition formula for the weighted zeta function of an irregular covering of a graph by its weighted L-functions. Moreover, we give a factorization of the weighted zeta function of an (irregular or regular) covering of a graph by equivalence classes of prime, reduced cycles of the base graph. As an application, we discuss the structure of balanced coverings of signed graphs.

In this papel; we present eficient parallel algorithms forjinding a minimum weighted connected dominating set, a minimum weighted Steiner tree for a distance-hereditary graph which take O(1og n) time using O(n+m) processors on a CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively. We also find a maximum weighted clique of a distance-hereditary graph in O...

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specifie...

We define a weighted zeta function of a digraph and a weighted L-function of a symmetric digraph, and give determinant expressions of them. Furthermore, we give a decomposition formula for the weighted zeta function of a g-cyclic -cover of a symmetric digraph for any finite group and g ∈ . A decomposition formula for the weighted zeta function of an oriented line graph L(G̃) of a regular coverin...

Automatic differentiation is a technique for the rule-based transformation of a subprogram that computes some mathematical function into a subprogram that computes the derivatives of that function. Automatic differentiation algorithms are typically expressed as operating on a weighted term graph called a linearized computational graph. Constructing this weighted term graph for imperative progra...