Core percolation in random graphs: a critical phenomena analysis

We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the "core". In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at "alpha = e = 2.718...": below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs. Key words: random graphs, leaf removal, core percolation, critical exponents, combinatorial optimization, finite size scaling, Monte-Carlo.Comment: 15 pages, 9 figures (color eps) [v2: published text with a new Title and addition of an appendix, a ref. and a fig.

Orthogonal Graph Drawing with Inflexible Edges

We consider the problem of creating plane orthogonal drawings of 4-planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge $e$ a natural number $\mathrm{flex}(e)$, its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge $e$ has at most $\mathrm{flex}(e)$ bends. It is known that FlexDraw is NP-hard if $\mathrm{flex}(e) = 0$ for every edge $e$. On the other hand, FlexDraw can be solved efficiently if $\mathrm{flex}(e) \ge 1$ and is trivial if $\mathrm{flex}(e) \ge 2$ for every edge $e$. To close the gap between the NP-hardness for $\mathrm{flex}(e) = 0$ and the efficient algorithm for $\mathrm{flex}(e) \ge 1$, we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility~$0$). We show that for any $\varepsilon > 0$ FlexDraw is NP-complete for instances with $O(n^\varepsilon)$ inflexible edges with pairwise distance $\Omega(n^{1-\varepsilon})$ (including the case where they induce a matching). On the other hand, we give an FPT-algorithm with running time $O(2^k\cdot n \cdot T_{\mathrm{flow}}(n))$, where $T_{\mathrm{flow}}(n)$ is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and $k$ is the number of inflexible edges having at least one endpoint of degree 4.Comment: 23 pages, 5 figure