Characterization of color patterns by dynamic [formula omitted]-paths

Abstract

Let H be a digraph possibly with loops, let D be a multidigraph, and let c:A(D)→V(H) be a coloring of the arcs of D with the vertices of H. A sequence W of vertices in D, W=(x0,x1,…,xn), is a dynamic H-walk if for each i∈{0,…,n−2} there exist fi and fi+1 such that fi is an arc from xi to xi+1, fi+1 is an arc from xi+1 to xi+2, and (c(fi),c(fi+1)) is an arc of H. If W does not repeat vertices, W is a dynamic H-path. For u,v∈V(D), we say that u reaches v by dynamic H-paths if there exists a dynamic H-path from u to v in D. A subset K⊆V(D) is a kernel by dynamic H-paths of D if every vertex in V(D)−K reaches some vertex in K by dynamic H-paths (absorbent by dynamic H-paths), and no vertex in K can be reached by another vertex in K by dynamic H-paths in D (independent by dynamic H-paths).Let D̃3 be the family of digraphs H such that for every H-arc-colored multidigraph D, D has a kernel by dynamic H-paths. Let D̃2 (resp. D̃1) be the class of digraphs H such that for every H-arc-colored multidigraph D (resp. multitournament), D has an independent absorbent by dynamic H-paths set. In this work, we provide a characterization of D̃2 and show that D̃3 is one of two families.