Characterising graph classes by forbidding a set of graphs has been a common feature in graph theory for many years. Up to date many classes have been char-acterised with respect to the induced subgraph relation and there are a plethora of results regarding graph classes closed under the minor relation. We consider the characterisation of graph classes closed under partial orders including edge...

For each minor-closed graph class we show that a simple variant of Borůvka’s algorithm computes a MST for any input graph belonging to that class with linear costs. Among minor-closed graph classes are e.g planar graphs, graphs of bounded genus, partial k-trees for fixed k, and linkless or knotless embedable graphs. The algorithm can be implemented on a CRCW PRAM to run in logarithmic time with...

this paper deals with leader-following and leaderless consensus problems of high-order multi-input/multi-output (mimo) multi-agent systems with unknown nonlinear dynamics in the presence of uncertain external disturbances. the agents may have different dynamics and communicate together under a directed graph. a distributed adaptive method is designed for both cases. the structures of the contro...

25 The Traveling Salesman Problem (TSP) is still one of the most researched topics in computational mathematics, and we introduce a variant of it, namely the study of the closed k-walks in graphs. We 2 G. Bullington, R. Gera, L. Eroh And S.J. Winters search for a shortest closed route visiting k cities in a non complete graph without weights. This motivates the following definition. Given a set...

Ω is weak-geodetically closed whenever Ω is weak-geodetically closed with respect to w for any w ∈ X. ∗A manuscript for algebraic combinatorics workshop 7/2-4 in National Hsinchu University of Education, an extension of 2007 International Conference on Graph Theory and Combinatorics & Fourth Cross-strait Conference on Graph Theory and Combinatorics. †Department of Applied Mathematics, National ...

The family of regular closed subsets of a topological space is used to introduce two concepts concerning a function f from a space X to a space Y . The first of them is the notion of f being rc-continuous. One of the established results states that a space Y is extremally disconnected if and only if each continuous function from a space X to Y is rc-continuous. The second concept studied is the...

Using graph-theoretical techniques, we establish an inequality regarding the number of walks and closed walks in a graph. This inequality yields several upper bounds for the number of closed walks in a graph in terms of the number of vertices, number of edges, maximum degree, degree sequence, and the Zagreb indices of the graph. As applications, we also present some new upper bounds on the Estr...

Two 2-cell embeddings ı : X → S and  : X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h : S → S and a graph automorphism γ of X such that ıh = γ. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of 2-cell embeddings of a simple graph (w...