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We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.

We introduce the general notion of structure resolvability and structure irresolvability, generalizing the usual concepts of resolvability and irresolvability.

The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{\alpha }}_{\alpha \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $dim\left(X_{\alpha }\cap X_{\beta }\right)=n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $cl{c}_{\mathbb{Z}}^{n+1}$, where n ≥ 1, and G is an Abelian group. Let ${X_{\alpha }}_{\alpha \in J}$ be an uncountable family of closed subsets of X. If $di{m}_{G}X=di{m}_G{X}_{\alpha }=n$ for all α ∈ J, then $di{m}_G\left(X_{\alpha }\cap X_{\beta }\right)=n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...