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...

We construct a quantum searching model of signed edge driven by walk. The time evolution operator this walk provides weighted adjacency matrix induced the assignment sign to each edge. This can be regarded as so-called coloring. Then an application, under arbitrary coloring which gives matching on complete graph $$n+1$$ vertices we consider search colored from set graph. show that finds within...

In this paper we employ Tutte’s theory of bridges to derive a decomposition theorem for binary matroids arising from signed graphs. The proposed decomposition differs from previous decomposition results on matroids that have appeared in the literature in the sense that it is not based on k-sums, but rather on the operation of deletion of a cocircuit. Specifically, it is shown that certain minor...

7 We study parallel complexity of signed graphs motivated by the highly 8 complex genetic recombination processes in ciliates. The molecular gene 9 assembly operations have been modeled by operations of signed graphs, 10 i.e., graphs where the edges have a sign + or −. In the optimization prob11 lem for signed graphs one wishes to find the parallel complexity by which 12 the graph can be reduce...