Acyclicity in edge-colored graphs

Gregory Z. Gutin, Mark Jones, Bin Sheng, Magnus Wahlström, Anders Yeo

Abstract: A walk W<math><mi is="true">W</mi></math> in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in W<math><mi is="true">W</mi></math> 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 i<math><mi is="true">i</mi></math> is a proper superset of graphs of acyclicity of type i+1<math><mi is="true">i</mi><mo is="true">+</mo><mn is="true">1</mn></math>, i=1,2,3,4.<math><mi is="true">i</mi><mo is="true">=</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">3</mn><mo is="true">,</mo><mn is="true">4</mn><mo is="true">.</mo></math> The first three types are equivalent to the absence of PC cycles, PC closed trails, and PC closed 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.