Paths and trails in edge-colored graphs

Abstract

a b s t r a c tThis paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s and t in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s − t path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s and t for a particular class of graphs and characterize edge-colored graphs without properly edgecolored closed trails. Next, we prove that deciding whether there exist k pairwise vertex/edge disjoint properly edge-colored s − t paths/trails in a c-edge-colored graph G c is NP-complete even for k = 2 and c = Ω(n 2 ), where n denotes the number of vertices in G c . Moreover, we prove that these problems remain NP-complete for c-edgecolored graphs containing no properly edge-colored cycles and c = Ω(n). We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.

Complexity of Paths, Trails and Circuits in Arc-Colored Digraphs

“…Problems regarding properly edge-colored paths, trails and cycles (or pec paths, trails and cycles, for short) in c-edge-colored (undirected) graphs have been widely studied from a graph theory and algorithmic point of views (see [3,1,24], the book [5] and the recent survey [18]). For instance, in [23], the author gives polynomial algorithms for several problems, including the determination of a pec s-t path (if one exists).…”

“…For instance, in [23], the author gives polynomial algorithms for several problems, including the determination of a pec s-t path (if one exists). More recently, the authors in [1] introduced the notion of trail-path graph. Using this concept, they extend Szeider's Algorithm to deal with pec s-t trails and they propose a polynomial algorithm for the determination of a pec s-t trail.…”

Joint routing and channel assignment in multi-channel wireless infrastructure networks

“…The problem of computing minimum cost "properly edgecolored path" (same as the minimum cost CDC-path) between two given nodes in a network where every link has at most one color (channel) available is addressed in [28]. For multichannel wireless networks considered in this paper, a link may have multiple channels available.…”

“…A CDC-path may be computed by assigning any one of the available channels on each of the links. We extend the Edmonds-Szeider (ES) node expansion employed in [28] to multi-channel wireless networks, as described below.…”

Acyclicity in edge-colored graphs

“…We will see that graphs of the first two types coincide with edge-colored graphs without PC cycles and without PC closed trails, respectively. These two classes of edge-colored graphs were characterized by Yeo [20] and Abouelaoualim et al [1], respectively. We will prove that graphs of PC acyclicity of type 3 are edge-colored graphs without PC walks.…”