3 research outputs found
Acyclicity in edge-colored graphs
A walk in edge-colored graphs is called properly colored (PC) if every
pair of consecutive edges in is of different color. We introduce and study
five types of PC acyclicity in edge-colored graphs such that graphs of PC
acyclicity of type is a proper superset of graphs of acyclicity of type
, The first three types are equivalent to the absence of PC
cycles, PC trails, and PC walks, respectively. While graphs of types 1, 2 and 3
can be recognized in polynomial time, the problem of recognizing graphs of type
4 is, somewhat surprisingly, NP-hard even for 2-edge-colored graphs (i.e., when
only two colors are used). The same problem with respect to type 5 is
polynomial-time solvable for all edge-colored graphs. Using the five types, we
investigate the border between intractability and tractability for the problems
of finding the maximum number of internally vertex disjoint PC paths between
two vertices and the minimum number of vertices to meet all PC paths between
two vertices