Abstract
Graph Theory In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.
Highlights
In the last few years a great number of applications have been modelled as problems in edge-colored graphs
We show how to polynomially solve this problem over graphs containing no PEC closed trails and prove that it becomes NP-complete over graphs containing no PEC cycles
We prove that the (3,B2)-SAT problem can be reduced to the problem of finding k = m + 2n vertex disjoint PEC s-t paths with length at most L + 2 in Gc, where Gc has no PEC cycles through s or t
Summary
In the last few years a great number of applications have been modelled as problems in edge-colored graphs. A PEC path or trail in Gc is said to be closed if its endpoints coincide and its first and last edges differ in color. They are refereed, respectively, as PEC cycle and PEC closed trail. K is an almost PEC cycle (resp., closed trail) through x in Gc if c(xv1) = c(xvk) and both paths (resp., trails) with at least two edges from x to v1 and from x to vk are properly edge-colored. The subgraph Fk denotes an union of k PEC paths and an arbitrarily number of PEC cycles in Gc, all vertex disjoint.
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