On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor

The generalised colouring numbers $\mathrm{col}_r(G)$ and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the $r$-colouring number $\mathrm{col}_r$ and a polynomial bound for the weak $r$-colouring number $\mathrm{wcol}_r$. In particular, we show that if $G$ excludes $K_t$ as a minor, for some fixed $t\ge4$, then $\mathrm{col}_r(G)\le\binom{t-1}{2}\,(2r+1)$ and $\mathrm{wcol}_r(G)\le\binom{r+t-2}{t-2}\cdot(t-3)(2r+1)\in\mathcal{O}(r^{\,t-1})$. In the case of graphs $G$ of bounded genus $g$, we improve the bounds to $\mathrm{col}_r(G)\le(2g+3)(2r+1)$ (and even $\mathrm{col}_r(G)\le5r+1$ if $g=0$, i.e. if $G$ is planar) and $\mathrm{wcol}_r(G)\le\Bigl(2g+\binom{r+2}{2}\Bigr)\,(2r+1)$.Comment: 21 pages, to appear in European Journal of Combinatoric

Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull-number of a graph and show that computing it is $\Sigma^P_2$ complete.Comment: 13 pages, 3 figure

Burning a Graph is Hard

Graph burning is a model for the spread of social contagion. The burning number is a graph parameter associated with graph burning that measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We prove that the corresponding graph decision problem is \textbf{NP}-complete when restricted to acyclic graphs with maximum degree three, spider graphs and path-forests. We provide polynomial time algorithms for finding the burning number of spider graphs and path-forests if the number of arms and components, respectively, are fixed.Comment: 20 Pages, 4 figures, presented at GRASTA-MAC 2015 (October 19-23rd, 2015, Montr\'eal, Canada