Abstract In this paper the class of Brauer graph algebras is proved to be closed under derived equivalence. For that we use rank maximal torus identity component $$Out^0(A)$$ O u t 0 ( A )</mml:m...

The problem of arbitrary decomposition of a graph G into closed trails i.e. a decomposition into closed trails of prescribed lengths summing up to the size of the graph G was first considered in the case of the complete graph G = Kn (for odd n) in connection with vertexdistinguishing coloring of the union of cycles. Next, the same problem was investigated for other families of graphs. In this p...

Abstract For a graph G , two dominating sets D and $$D'$$ D ′ in non-negative integer k the set is said to -transform if there sequence $$D_0,\ldots ,D_\ell $$ 0 , … ℓ ...

For an arbitrary subset P of the reals, we define a function f :V → P to be a Pdominating function of graph G = (V,E) if the sum of the function values over any closed neighbourhood is at least 1. That is, for every v ∈ V , f(N(v) ∪ {v}) ≥ 1. The P-domination number of a graph is defined to be the infimum of f(V ) taken over all P-dominating functions f . When P = {0, 1} one obtains the standar...

Introduction LET dim-X be the covering dimension of a space X and let ind X and Ind X be the dimensions defined inductively in terms of the boundaries of neighbourhoods of points and closed sets respectively. The local dimension loo dim X is the least number n such that every point has a closed neighbourhood U with dim U ^ n. The local inductive dimension locIndX is defined analogously, while i...

Some graph-theoretical tools are used to introduce the concept of "regular genus" §(M„), for every closed n-dimensional PL-manifold M„. Then it is proved that the regular genus of every surface equals its genus, and that the regular genus of every 3-manifold Af j equals its Heegaard genus, if M3 is orientable, and twice its Heegaard genus, if M3 is nonorientable. A geometric approach, and some ...

The closed neighbourhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end vertex with e. Let f be a function on the edges of G into the set {−1, 1}. If e∈NG[x] f(e) ≥ 1 for every x ∈ E(G), then f is called a signed edge domination function of G. The minimum value of ∑ x∈E(G) f(x), taken over every signed edge domination function f of G, is called s...