An oriented walk double covering of a graph G is a set of oriented closed walks which, traversed successively, combined will have traced each edge of G once in each direction. A bidirectional double tracing of a graph G is an oriented walk double covering which consists of a single closed walk. A retracting in a closed walk is the immediate succession of an edge by its inverse. Every graph with...

We characterize the edge-signed graphs in which every 'significant' positive closed walk (or combination of walks) has even length, under seven different criteria for significance, and also those edge-signed graphs whose double covering graph is bipartite. If the property of even length is generalized to positivity in a second edge signing, the characterizations generalize as well. We also char...

We shall prove that a connected graph G is projective-planar if and only if it has a 2n-fold planar connected covering obtained as a composition of an n-fold covering and a double covering for some n¿ 1 and show that every planar regular covering of a nonplanar graph is such a composite covering. c © 2002 Elsevier Science B.V. All rights reserved.

An outer-independent double Italian dominating function (OIDIDF)on a graph $G$ with vertex set $V(G)$ is a function$f:V(G)longrightarrow {0,1,2,3}$ such that if $f(v)in{0,1}$ for a vertex $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$,and the set $ {uin V(G)|f(u)=0}$ is independent. The weight ofan OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. Theminimum weight of an OIDIDF on a graph $G$ is cal...

This paper addresses a drayage problem, which is motivated by the case study of a real carrier. Its trucks carry one or two containers from a port to importers and from exporters to the port. Since up to four customers can be served in each route, we propose a set-covering formulation for this problem where all possible routes are enumerated. This model can be efficiently solved to optimality b...