Algorithms to Measure Diversity and Clustering in Social Networks through Dot Product Graphs

Social networks are often analyzed through a graph model of the network. The dot product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d-dimensional vector a v represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if a u · a v  ≥ t, for some fixed, positive threshold t. The resulting graph is called a d-dot product graph.. We consider two measures for diversity and clustering in social networks by using a d-dot product graph model for the network. Diversity is measured through the size of the largest independent set of the graph, and clustering is measured through the size of the largest clique. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d = 2, but NP-complete for d ≥ 3. We show that the clustering problem is polynomial-time solvable for d = 2. To our knowledge, these results are also the first on the computational complexity of combinatorial optimization problems on dot product graphs. We also consider the situation when two individuals are connected if their preferences are not opposite. This leads to a variant of the standard dot product graph model by taking the threshold t to be zero. We prove in this case that the diversity problem is polynomial-time solvable for any fixed d

Parameterized complexity of vertex colouring

AbstractFor a family F of graphs and a nonnegative integer k, F+ke and F−ke, respectively, denote the families of graphs that can be obtained from F graphs by adding and deleting at most k edges, and F+kv denotes the family of graphs that can be made into F graphs by deleting at most k vertices.This paper is mainly concerned with the parameterized complexity of the vertex colouring problem on F+ke, F−ke and F+kv for various families F of graphs. In particular, it is shown that the vertex colouring problem is fixed-parameter tractable (linear time for each fixed k) for split+ke graphs and split−ke graphs, solvable in polynomial time for each fixed k but W[1]-hard for split+kv graphs. Furthermore, the problem is solvable in linear time for bipartite+1v graphs and bipartite+2e graphs but, surprisingly, NP-complete for bipartite+2v graphs and bipartite+3e graphs

On Spanning 2-Trees in a Graph

A k-tree is either a complete graph on k vertices or a graph T that contains a vertex whose neighbourhood in T induces a complete graph on k vertices and whose removal results in a k-tree. A subgraph of a graph is a spanning k-tree if it is a k-tree and contains every vertex of the graph. This paper is concerned with spanning 2-trees in a graph. It is shown that spanning 2-trees have close connections with two special types of spanning trees: locally-connected spanning trees (A locally-connected spanning tree of a graph G is a spanning tree such that for every vertex v of T the neighbourhood of v in T induces a connected subgraph in G) and tree 2-spanners (A tree 2-spanner of a graph G is a spanning tree such that for every edge of G not in T the distance in T between the two ends of the edge is two). An approximation algorithm is presented for finding a minimum-weight spanning 2-tree in a weighted complete graph, whose asymptotic performance ratio is at most 2 when edge w..