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...

A graph with signed edges is orientation embedded in a surface when it is topologically embedded so that one trip around a closed path preserves or reverses orientation according as the path's sign product is positive or negative. We nd the smallest surface within which it is possible to orientation-embed the complete bipartite signed graph K r;s , which is obtained from the complete bipartite ...

A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a far-reaching generalization of this, conjectured by Wagner: If a class of graphs is minor-closed (i.e., it is...

In the final paper of the Graph Minors series N. Robertson and P. Seymour proved that graphs are well-quasi-ordered under the immersion ordering. A direct implication of this theorem is that each class of graphs that is closed under taking immersions can be fully characterized by forbidding a finite set of graphs (immersion obstruction set). However, as the proof of the well-quasi-ordering theo...