In this paper we study an open problem: Is it true that each homotopy domination of a polyhedron over itself is equivalence? Since every ANR has the type polyhedron, question related in obvious way to longstanding Borsuk's problem (1967) two ANR's dominating other have same type? The answer positive for all polyhedra (or ANR's) with polycyclic-by-finite fundamental groups, manifolds, and clearl...

Abstract We introduce the concept of a homotopy motion subset in manifold and give systematic study motions surfaces closed orientable 3-manifolds. This notion arises from various natural problems 3-manifold theory such as domination pairs, homotopical behavior simple loops on Heegaard surface monodromies virtual branched covering bundles associated with splitting.

For r≥1, the r-independence complex of a graph G, denoted Indr(G), is simplicial whose faces are subsets I⊆V(G) such that each component induced subgraph G[I] has at most r vertices. In this article, we establish relation between distance r-domination number G and (homological) connectivity Indr(G). We also prove for chordal either contractible or homotopy equivalent to wedge spheres. Given sph...

Let H be a hypergraph on finite set V. A cover of is vertices that meets all edges H. If W not H, then said to noncover The complex the abstract simplicial whose faces are noncovers In this paper, we study homological properties complexes hypergraphs. particular, obtain an upper bound their Leray numbers. in terms domination Also, our proof idea applied compute homotopy type certain uniform hyp...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...

In the recent years several complexes arising from forests have been studied by different authors with different techniques (see [BM], [BLN], [EH], [E], [Kl], [K1], [K2], [MT], [W]). The interest in these problems is motivated by applications in different contexts, such as graph theory and statistical mechanics ([BK], [BLN], [J]). We introduce a unifying approach to study the homotopy type of s...

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique c...