On Markov graphs

We investigate graph-theoretical properties of Markov graphs from dynamical systems

On disjoint union of M-graphs

Given a pair (X,σ) consisting of a finite tree X and its vertex self-map σ one can construct the corresponding Markov graph Γ(X,σ) which is a digraph that encodes σ-covering relation between edges in X. M-graphs are Markov graphs up to isomorphism. We obtain several sufficient conditions for the disjoint union of M-graphs to be an M-graph and prove that each weak component of M-graph is an M-graph itself

On the hardness of switching to a small number of edges

Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other one by a sequence of switches. Jel\'inkov\'a et al. [DMTCS 13, no. 2, 2011] presented a proof that it is NP-complete to decide if the input graph can be switched to contain at most a given number of edges. There turns out to be a flaw in their proof. We present a correct proof. Furthermore, we prove that the problem remains NP-complete even when restricted to graphs whose density is bounded from above by an arbitrary fixed constant. This partially answers a question of Matou\v{s}ek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014].Comment: 19 pages, 7 figures. An extended abstract submitted to COCOON 201

On graphs with graphic imbalance sequences

The imbalance of the edge e = uv in a graph G is the value imbG(e) = |dG(u) − dG(v)|. We prove that the sequence MG of all edge imbalances in G is graphic for several classes of graphs including trees, graphs in which all non-leaf vertices form a clique and the so-called complete extensions of paths, cycles and complete graphs. Also, we formulate two interesting conjectures related to graphicality of MG