Efficient Algorithms for Node Disjoint Subgraph Homeomorphism Determination

Recently, great efforts have been dedicated to researches on the management of large scale graph based data such as WWW, social networks, biological networks. In the study of graph based data management, node disjoint subgraph homeomorphism relation between graphs is more suitable than (sub)graph isomorphism in many cases, especially in those cases that node skipping and node mismatching are allowed. However, no efficient node disjoint subgraph homeomorphism determination (ndSHD) algorithms have been available. In this paper, we propose two computationally efficient ndSHD algorithms based on state spaces searching with backtracking, which employ many heuristics to prune the search spaces. Experimental results on synthetic data sets show that the proposed algorithms are efficient, require relative little time in most of the testing cases, can scale to large or dense graphs, and can accommodate to more complex fuzzy matching cases.Comment: 15 pages, 11 figures, submitted to DASFAA 200

Slider-pinning Rigidity: a Maxwell-Laman-type Theorem

We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley's direction networks.Comment: Accepted, to appear in Discrete and Computational Geometr

A new graph parameter related to bounded rank positive semidefinite matrix completions

The Gram dimension \gd(G) of a graph $G$ is the smallest integer $k\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$, can be completed to a positive semidefinite matrix of rank at most $k$ (assuming a positive semidefinite completion exists). For any fixed $k$ the class of graphs satisfying \gd(G) \le k is minor closed, hence it can characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is $K_{k+1}$ for $k\le 3$ and that there are two minimal forbidden minors: $K_5$ and $K_{2,2,2}$ for $k=4$. We also show some close connections to Euclidean realizations of graphs and to the graph parameter $\nu^=(G)$ of \cite{H03}. In particular, our characterization of the graphs with \gd(G)\le 4 implies the forbidden minor characterization of the 3-realizable graphs of Belk and Connelly \cite{Belk,BC} and of the graphs with $\nu^=(G) \le 4$ of van der Holst \cite{H03}.Comment: 31 pages, 6 Figures. arXiv admin note: substantial text overlap with arXiv:1112.596