New recursive constructions of amoebas and their balancing number

The definition of amoeba graphs is based on iterative \emph{feasible edge-replacements}, where, at each step, an edge from the graph is removed and placed in an available spot in a way that the resulting graph is isomorphic to the original graph. Broadly speaking, amoebas are graphs that, by means of a chain of feasible edge-replacements, can be transformed into any other copy of itself on a given vertex set (which is defined according to whether these are local or global amoebas). Global amoebas were born as examples of \emph{balanceable} graphs, which are graphs that appear with half of their edges in each color in any $2$-edge coloring of a large enough complete graph with a sufficient amount of edges in each color. The least amount of edges required in each color is called the \emph{balancing number} of $G$. In a work by Caro et al., an infinite family of global amoeba trees with arbitrarily large maximum degree is presented, and the question if they were also local amoebas is raised. In this paper, we provide a recursive construction to generate very diverse infinite families of local and global amoebas, by which not only this question is answered positively, it also yields an efficient algorithm that, given any copy of the graph on the same vertex set, provides a chain of feasible edge-replacements that one can perform in order to move the graph into the aimed copy. All results are illustrated by applying them to three different families of local amoebas, including the Fibonacci-type trees. Concerning the balancing number of a global amoeba $G$, we are able to express it in terms of the extremal number of a class of subgraphs of $G$. By means of this, we give a general lower bound for the balancing number of a global amoeba $G$, and we provide linear (in terms of order) lower and upper bounds for the balancing number of our three case studies.Comment: 27 pages, 10 figure

The VC-Dimension of Graphs with Respect to k-Connected Subgraphs

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width

Decomposition of bounded degree graphs into C4C_4-free subgraphs

We prove that every graph with maximum degree $\Delta$ admits a partition of its edges into $O(\sqrt{\Delta})$ parts (as $\Delta\to\infty$) none of which contains $C_4$ as a subgraph. This bound is sharp up to a constant factor. Our proof uses an iterated random colouring procedure.Comment: 8 pages; to appear in European Journal of Combinatoric

Planar Induced Subgraphs of Sparse Graphs

We show that every graph has an induced pseudoforest of at least $n-m/4.5$ vertices, an induced partial 2-tree of at least $n-m/5$ vertices, and an induced planar subgraph of at least $n-m/5.2174$ vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest $K_h$-minor-free graph in a given graph can sometimes be at most $n-m/6+o(m)$.Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph Algorithms and Application

Network synchronizability analysis: the theory of subgraphs and complementary graphs

In this paper, subgraphs and complementary graphs are used to analyze the network synchronizability. Some sharp and attainable bounds are provided for the eigenratio of the network structural matrix, which characterizes the network synchronizability, especially when the network's corresponding graph has cycles, chains, bipartite graphs or product graphs as its subgraphs.Comment: 13 pages, 7 figure