The First Order Definability of Graphs with Separators via the Ehrenfeucht Game

We say that a first order formula $\Phi$ defines a graph $G$ if $\Phi$ is true on $G$ and false on every graph $G'$ non-isomorphic with $G$. Let $D(G)$ be the minimal quantifier rank of a such formula. We prove that, if $G$ is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then $D(G)=O(\log n)$, where $n$ denotes the order of $G$. This bound is optimal up to a constant factor. If $h$ is a constant, for connected graphs with no minor $K_h$ and degree $O(\sqrt n/\log n)$, we prove the bound $D(G)=O(\sqrt n)$. This result applies to planar graphs and, more generally, to graphs of bounded genus.Comment: 17 page

On the Obfuscation Complexity of Planar Graphs

Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity of $G$, to be the maximum number of edge crossings in a drawing of $G$. Relating $obf(G)$ to the distribution of vertex degrees in $G$, we show an efficient way of constructing a drawing of $G$ with at least $obf(G)/3$ edge crossings. We prove bounds (\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2 for an $n$-vertex planar graph $G$ with minimum vertex degree $\delta(G)\ge 2$. The shift complexity of $G$, denoted by $shift(G)$, is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of $G$ (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If $\delta(G)\ge 3$, then $shift(G)$ is linear in the number of vertices due to the known fact that the matching number of $G$ is linear. However, in the case $\delta(G)\ge2$ we notice that $shift(G)$ can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing $D$ of a planar graph, it is NP-hard to find an optimum sequence of shifts making $D$ crossing-free.Comment: 12 pages, 1 figure. The proof of Theorem 3 is simplified. An overview of a related work is adde