We consider a generalized Path Traveling Salesman Problem where the distances are defined by a 2-edge-connected graph metric and a constant number of salesmen have to cover all the destinations by traveling along paths of minimum total length. We show that for this problem there is a polynomial algorithm with asymptotic approximation ratio of 2 .

Given a tree and a set of paths in the tree, the problem of finding a minimum number of paths from the given path set to cover all the vertices in the tree is investigated in the paper. To distinguish from the classical path cover problem, such an optimization problem is referred to as vertex covering by paths. The problem and its edge variant, edge covering by paths, find applications in machi...

The k-th semi total point graph of a graph G, , ‎is a graph‎ obtained from G by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎In this paper‎, a formula for Laplacian polynomial of in terms of‎ characteristic and Laplacian polynomials of G is computed‎, ‎where is a connected regular graph‎.The Kirchhoff index of is also computed‎.

Modern, inherently dynamic systems are usually characterized by a network structure, i.e. an underlying graph topology, which is subject to discrete changes over time. Given a static underlying graph G, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge of G, indicating the discrete time steps when this edge is active. While most of the recent th...

The cover polynomial C(D) = C(D;x, y) of a digraph D is a twovariable polynomial whose coefficients are determined by the number of vertex coverings of D by directed paths and cycles. Just as for the Tutte polynomial for undirected graphs (cf. [11, 16]), various properties of D can be read off from the values of C(D;x, y). For example, for an n-vertex digraph D, C(D; 1, 0) is the number of Hami...

A cycle in a matroid is a disjoint union of circuits. This paper proves that every regular matroid M without coloops has a set S of cycles whose union is E(M) such that every element is in at most 3 of the cycles in S. It follows immediately from this that, on average, each element of M is in at most 3 members of the cycle cover S. This improves on a 1989 result of Jamshy and Tarsi who proved t...

Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise Teichmuller fibred face Thurston norm ball. We prove that triangulation is equal to certain twisted Alexander underlying manifold. Then we give formulas relating untwisted polynomial. There are two formulas; one holds when maximal free abelian cover edge-orientable, another it not ed...

An edge cover C of an undirected graph is a set of edges such that every vertex has an adjacent edge in C. We show that a Glauber dynamics Markov chain for edge covers mixes rapidly for graphs with degrees at most three. Glauber dynamics have been studied extensively in the statistical physics community, with special emphasis on lattice graphs. Our results apply, for example, to the hexagonal l...

Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and b are positive integers. A function is said to be a signed -edge cover of G if G ( ) V G ( ) e E v ( ) E G G : ( f E k b k ) { 1,1} G   ( , ) b k ( ) f e b    for at least vertices of , where . The value k v G ( ) = {uv E( ( ) E v G u N v   ) | } ( ) min ( ) G e E f e   , taki...

The following is proved: if every bridgeless graph G has a cycle cover of length at most 7/51€(G)I, then every bridgeless graph G has a cycle cover of length at most 7/51€(G)I such that any edge of G is covered once or twice.