We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, ...

We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable if the maximum degree of the input graph is at most 3 (respectively 4), whereas it is APX-hard for general graphs and NP-hard for planar graphs if the maximum degree is 4 (respectiv...

In general the problem of finding a miminum spanning tree for a weighted directed graph is difficult but solvable. There are a lot of differences between problems for directed and undirected graphs, therefore the algorithms for undirected graphs cannot usually be applied to the directed case. In this paper we examine the kind of weights such that the problems are equivalent and a minimum spanni...

Any graph problem, which is NP-hard in general graphs, becomes polynomial-time solvable when restricted to graphs in special classes. When does a difficult problem become easy? To answer this question we study the notion of boundary classes. In the present paper we define this notion in its most general form, describe several approaches to identify boundary classes and apply them to various gra...

in this paper, we classify the indecomposable non-nilpotent solvable lie algebras with $n(r_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $n(r_n,m,r)$.we also prove that these solvable lie algebras are complete and unique, up to isomorphism.

applications of hypergroups have mainly appeared in special subclasses. one of the important subclasses is the class of polygroups. in this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. in particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.

An efficiently total dominating set of a graph G is a subset of its vertices such that each vertex of G is adjacent to exactly one vertex of the subset. If there is such a subset, then G is an efficiently total dominable graph (G is etd). In this paper, we prove NP-completeness of the etd decision problem on the class of planar bipartite graphs of maximum degree 3. Furthermore, we give an effic...

We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, languages their automata context-free. On other hand, we show there is monoid with hyperbolic and an unsolvable problem.

In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...

For a connected graph G=(V,E), matching M⊆E is cut of G if G−M disconnected. It known that for an integer d, the corresponding decision problem Matching Cut polynomial-time solvable graphs diameter at most d d≤2 and NP-complete d≥3. We prove same dichotomy bounded radius. H, H-free it does not contain H as induced subgraph. As consequence our result, we can solve in polynomial time P6-free grap...