The control flow of programs can be represented by directed graphs. In this paper we provide a uniform and detailed formal basis for control flow graphs combining known definitions and results with new aspects. Two graph reductions are defined using only syntactical information about the graphs, but no semantical information about the represented programs. We prove some properties of reduced graphs and also about the paths in reduced graphs. Based on graphs, we define statement coverage and branch...

The join of two graphs $G$ and $H$ is a graph formed from disjoint copies of $G$ and $H$ by connecting each vertex of $G$ to each vertex of $H$. We determine the flow number of the resulting graph. More precisely, we prove that the join of two graphs admits a nowhere-zero $3$-flow except for a few classes of graphs: a single vertex joined with a graph containing an isolated vertex or an odd circuit tree component, a single edge joined with a graph containing only isolated edges, a single edge plus an isolated...

Let $G$ be a graph, and $\lambda$ the smallest integer for which $G$ has a nowhere-zero $\lambda$-flow, i.e., an integer $\lambda$ for which $G$ admits a nowhere-zero $\lambda$-flow, but it does not admit a $(\lambda -1)$-flow. We denote the minimum flow number of $G$ by $\Lambda \left(G\right)$. In this paper we show that if $G$ and $H$ are two arbitrary graphs and $G$ has no isolated vertex, then $\Lambda (G\vee H)\le 3$ except two cases: (i) One of the graphs $G$ and $H$ is $K_2$ and the other is $1$-regular. (ii) $H=K_1$ and $G$ is a graph with at least one isolated vertex or a component whose every block is an...

In the minimization of the number of subtours made by the insertion head of an SMD placement machine a variant of the network flow problem arose. In a network with $n$ vertices and $m$ arcs a set $F$ of arcs (parametrized arcs) is given. The task is to find a flow of a given size such that the maximum of flow values along the arcs from $F$ is minimized. This problem can be solved by a sequence of maximum flow computations in modified networks where the capacities of the parametrized arcs are successively...

We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M(G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M(G_{)}$. We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M(G_{)}$ admits a unique invariant...