A remark on homotopy and category domination.

A Remark on Chain-homotopy

lor every p-chain c* of the complex X. If ti~t2, then Ti~r2 (that is, ti, t2 induce identical homomorphisms from the homology groups of X into the homology groups of Y). It is well known that the converse is generally false. Several examples to this effect may be found in [2]. A very simple example is given in §2 below. On the other hand, if X is torsion-free, then the relation ti~t2 does imply...

A Remark on Total Domination Critical Graphs

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we disprove a conjecture posed in a recent paper(On an open problem concerning total domination critical graphs, Expo. Mat...

A remark on the (2, 2)-domination number

A subset D of the vertex set of a graph G is a (k, p)-dominating set if every vertex v ∈ V (G) \ D is within distance k to at least p vertices in D. The parameter γk,p(G) denotes the minimum cardinality of a (k, p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that γk,p(G) ≤ p p+k n(G) for any graph G with δk(G) ≥ k + p − 1, where the latter means that every vertex ...

Homotopy Coherent Category Theory

A Convenient Category for Directed Homotopy

We propose a convenient category for directed homotopy consisting of preordered topological spaces generated by cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable.