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

Optimal subgraph structures in scale-free configuration models

Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced subgraph. We obtain these results by analyzing the configuration model with degree exponent $\tau\in(2,3)$ and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize {\it all} subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the {\it erased} configuration model.Comment: 50 pages, 2 figure