H-Kernels in Infinite Digraphs

Abstract

Let H be a digraph possibly with loops and D a digraph (possibly infinite) without loops whose arcs are coloured with the vertices of H (D is an H-coloured digraph). V(D) and A(D) will denote the sets of vertices and arcs of D respectively. A directed walk or a directed path W in D is an H-walk or an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A set ${N \subseteq V(D}$) is an H-kernel if for every pair of different vertices in N there is no H-path between them, and for every vertex ${u \in V(D) \setminus N}$ there exists an H-path in D from u to N. Linek and Sands introduced the concept of H-walk and this concept was later used by several authors. In particular, Galeana-Sánchez and Delgado-Escalante used the concept of H-walk in order to introduce the concept of H-kernel, which generalizes the concepts of kernel and kernel by monochromatic paths. Let D be an arc-coloured digraph. In 2009 Galeana-Sánchez introduced the concept of color-class digraph of D, denoted by ${\fancyscript{C}_C(D),}$ as follows: the vertices of the color-class digraph are the colors represented in the arcs of D, and ${(i, j) \in A(\fancyscript{C}_C(D))}$ if and only if there exist two arcs namely (u, v) and (v, w) in D such that (u, v) has color i and (v, w) has color j. Since V ${(\fancyscript{C}_C(D)) \subseteq {\rm V}(H)}$ , the main question is: What structural properties of ${\fancyscript{C}_C(D),}$ with respect to H, imply that D has an H-kernel? Suppose that D has no infinite outward H-path. In this paper we prove that if ${\fancyscript{C}_C(D) \subseteq H}$ , then D has an H-kernel. We also prove that if there exists a partition (V ₁, V ₂) of V ${(\fancyscript{C}_C(D))}$ such that: (1) ${\fancyscript{C}_C(D)[V_i] \subseteq H[V_i]}$ for each i ${\in}$ {1,2}, (2) if ${(u, v) \in A(\fancyscript{C}_C(D))}$ for some ${u \in V_i}$ and for some ${v \in V_j}$ , with i ≠ j and ${i, j \in}$ {1,2}, then ${(u, v) \notin A(H),}$ and (3) D has no V _i -colored infinite outward H-path for each i ${\in}$ {1,2}. Then D has an H-kernel. Several previous results are generalized.