Excluding a Weakly 4-connected Minor

A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs

Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.Comment: 37 page

Weak degeneracy of graphs

Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every $d$-degenerate graph is also weakly $d$-degenerate. On the other hand, if $G$ is weakly $d$-degenerate, then $\chi(G) \leq d + 1$ (and, moreover, the same bound holds for the list-chromatic and even the DP-chromatic number of $G$). It turns out that several upper bounds in graph coloring theory can be phrased in terms of weak degeneracy. For example, we show that planar graphs are weakly $4$-degenerate, which implies Thomassen's famous theorem that planar graphs are $5$-list-colorable. We also prove a version of Brooks's theorem for weak degeneracy: a connected graph $G$ of maximum degree $d \geq 3$ is weakly $(d-1)$-degenerate unless $G \cong K_{d + 1}$. (By contrast, all $d$-regular graphs have degeneracy $d$.) We actually prove an even stronger result, namely that for every $d \geq 3$, there is $\epsilon > 0$ such that if $G$ is a graph of weak degeneracy at least $d$, then either $G$ contains a $(d+1)$-clique or the maximum average degree of $G$ is at least $d + \epsilon$. Finally, we show that graphs of maximum degree $d$ and either of girth at least $5$ or of bounded chromatic number are weakly $(d - \Omega(\sqrt{d}))$-degenerate, which is best possible up to the value of the implied constant.Comment: 21 p