Dismantlable Digraphs and the Fixed Point Property

In this paper, the concept of continuous multifunctions for digraphs is introduced. It turns out that a digraph multifunction is continuous if and only if it is strong. We introduce the notion of dismantlable digraphs and show that it does generalize the dismantlable graphs and posets. Generalizing various previous results, we show that any dismantlable digraph has the almost fixed point property (AFPP) for strong multifunctions. 1 Power digraph and digraph multifunctions By a directed graph (briefly digraph) D we mean a nonempty finite set V (D) of elements called points and a finite set A(G) of ordered pairs of distinct vertices called arcs (or arrows).. A point x of V (D) is said to be looped if (x, x) ∈ A(D); otherwise x is loopless . A digraph D is said to be reflexive (resp. irreflexive) if all its vertices are looped (resp. loopless). In this paper we concentrate solely on finite reflexive digraphs without multiple arcs. Note that it is clear that tolerance graphs and posets are digraphs. For digraphs C and D, a digraph homomorphism f : C → D maps points of C to points of D, preserving the arrows, i.e., (x, y) ∈ A(C) ⇒ (f(x), f(y)) ∈ A(D). By a digraph multifunction f : C → D we understand a mapping that assigns to a point x ∈ V (C) of C a non-empty subset f(x) ⊆ V (D) in D. Definition 1.1. Let D be a digraph. The strong power digraph Ps(D) of D has the non-empty finite subsets of V (D), P fin(V (D)) as points, and the arrow set As(D), satisfying (X,Y ) ∈ As(D) if and only if ∀x ∈ X, ∃y ∈ Y, (x, y) ∈ A(D) and ∀y ∈ Y, ∃x ∈ X, (x, y) ∈ A(D). Definition 1.2. The digraph multifunction f from C to D, f : C → D, is strong if f̂ : C → Ps(D), f̂(x) = f(x) ∈ P + fin(V (D)), for all x ∈ V (C), is a digraph homomorphism. Note that the strong digraph multifunctions reduce to digraph homomorphisms if they are singlevalued. And if f : C → D, g : D → E are strong digraph multifunctions, then the composition g◦f : C → E will be a strong digraph multifunction. Also in [8] the power constructions for tolerance graphs and simplicial complexes were developed, and, in terms of them, the above concept of multifunctions was extended to simplicial multifunctions. Let D be a digraph, it is clear that the strong power relation As(D) = A(Ps(D)) is the conjunction of the two “weak power relations” Al(D), Au(D) (in the notation of Brink [1]), defined by: (X,Y ) ∈ Al(D) ⇔ ∀x ∈ X, ∃y ∈ Y, (x, y) ∈ A(D); (X,Y ) ∈ Au(D) ⇔ ∀y ∈ Y, ∃x ∈ X, (x, y) ∈ A(D). Note that the relations Al and Au correspond respectively to the so-called “lower” (or Hoare) and “upper” (or Smyth) preorders in powerdomain theory. Of course, we have Al(D) = (Au(D)) −1