08191 Working Group Report – X-graphs of Y-graphs and their Representations

Abstract

We address graph decomposition problems that help the hybrid visualization of large
graphs, where different graphic metaphors (node-link, matrix, etc.) are used in the same
picture. We generalize the $X$-graphs of $Y$-graphs model introduced by Brandenburg
(Brandenburg, F.J.: Graph clustering I: Cycles of cliques. In Di Battista, G., 
ed.: Graph Drawing (Proc. GD '97). Volume 1353 of Lecture Notes Comput. Sci., Springer-Verlag
(1997) 158--168) to formalize the problem of automatically identifying dense subgraphs
($Y$-graphs, clusters) that are prone to be collapsed and shown with a matricial
representation when needed. We show that (planar, $K_5$)-recognition, that is, the
problem of identifying $K_5$ subgraphs such that the graph obtained by collapsing them
is planar, is NP-hard. On the positive side, we show that it is possible to determine the
highest value of $k$ such that $G$ is a (planar,$k$-core)-graph in $O(m + n log(n))$ time.