We consider a graph-based model for the study of parallelism in ciliate gene assembly, where a signed graph is associated to each micronuclear gene and the gene assembly is modeled as a graph rewriting process. A natural measure of complexity for gene assembly counts the minimal number of parallel steps needed to reduce the associated signed graph. We investigate the complexity of several class...

Let $G$ be a simple connected graph. In this paper, Szeged dimension and PI$_v$ dimension of graph $G$ are introduced. It is proved that if $G$ is a graph of Szeged dimension $1$ then line graph of $G$ is 2-connected. The dimensions of five composite graphs: sum, corona, composition, disjunction and symmetric difference with strongly regular components is computed. Also explicit formulas of Sze...

A signed graph is a pair (G,Σ) where G is a graph and Σ is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of Σ. A blocking pair of (G,Σ) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts i...

Does the interaction graph of a finite dynamical system can force this system to have a “complex” dynamics ? In other words, given a finite interval of integers A, which are the signed digraphsG such that every finite dynamical system f : An → An with G as interaction graph has a “complex” dynamics ? If |A| ≥ 3 we prove that no such signed digraph exists. More precisely, we prove that for every...

The goal of this work is to study homomorphism problems (from a computational point of view) on two superclasses of graphs: 2-edge-coloured graphs and signed graphs. On the one hand, we consider the H-Colouring problem when H is a 2-edge-coloured graph, and we show that a dichotomy theorem would imply the dichotomy conjecture of Feder and Vardi. On the other hand, we prove a dichotomy theorem f...

For the signed circuits of a regular oriented matroid (and more particular of a digraph) the above questions have been studied very well in past. From graph theory it is known that the dimension of the circuit space of a connected digraph is |E| − |V | + 1, that the circuit space L(C) is regular and that the elementary circuits {C(B, e)}e∈E\B form a basis of L(C) for any basis B of O. For gener...