Výpočetní složitost testování rovinnosti grafu

In this paper we will show that the problem of planarity testing is in SL (symmetric nondeterministic LOGSPACE). The main part of our proof is a reduction of the problem to planarity of graphs with maximal degree three. Note that usual replacing vertices of degree bigger than three by "little circles" can spoil planarity, we need to be smarter. Planarity of graphs with maximal degree three was already solved in paper "Symmetric complementation" by John Reif. Previously Meena Mahajan and Eric Allender have already proved this in ("Complexity of planarity testing"), but their proof is the pure SL implementation of a parallel algorithm by John Reif and Vijaya Ramachandran ("Planarity testing in parallel"). But it is possibly unnecessarily complex and sophisticated for the purposes of the space complexity. This result together with recent breakthrough by Omer Reingold that SL = L ("Undirected T-connectivity in log-space") completely solves the question of complexity of planarity problem, because planarity is hard for L (it is again shown in "Complexity of planarity testing"). We construct logarithmic-space computable function that converts input graph G into G0 with maximal degree three such that G is planar if and only if G0 is. This together with.V tomto článku ukážeme, že testování planarity je v SL (symetrický nedeterministický LOGSPACE). Hlavní část našeho důkazu je redukce na problém testování rovinnosti grafu s maximálním stupněm tři. Povšiměte si, že obvyklé nahrazování vrchol větších stupňů "malými kružnicemi" může rovinnost pokazit, musíme si počínat šikovněji. Testování rovinnosti grafu s maximálním stupněm tři už bylo vyřešeno ve článku "Symmetric complementation" Johna Reifa. Už dříve Meena Mahajan a Eric Allender ("Complexity of planarity testing") ukázali, že testování rovinnosti je v SL. Jejich důkaz se však sestává z SL implementace velmi složitého paralelního algoritmu od Johna Reifa a Vijayi Ramachandran ("Planarity testing in parallel"). Ten je však nejspíše zbytečně komplikovaný pro účely prostorové složitosti. Tento výsledek spolu s nedávným průlomem Omera Reingolda dokazujícího, že SL = L ("Undirected ST-connectivity in log-space") zcela řeší otázku složitosti testování planarity, protože to je těžké pro L (toto je též dokázáno v "Complexity of planarity testing"). Zkonstruujeme algoritmus používající logaritmický prostor, který převede vstupní graf G na G0 s maximálním stupněm 3 tak, že G je rovinný tehdy a jen tehdy, když G0 je rovinný.Katedra aplikované matematikyDepartment of Applied MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Beyond Outerplanarity

We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer $k$-planar graphs, where each edge is crossed by at most $k$ other edges; and, outer $k$-quasi-planar graphs where no $k$ edges can mutually cross. We show that the outer $k$-planar graphs are $(\lfloor\sqrt{4k+1}\rfloor+1)$-degenerate, and consequently that every outer $k$-planar graph can be $(\lfloor\sqrt{4k+1}\rfloor+2)$-colored, and this bound is tight. We further show that every outer $k$-planar graph has a balanced separator of size $O(k)$. This implies that every outer $k$-planar graph has treewidth $O(k)$. For fixed $k$, these small balanced separators allow us to obtain a simple quasi-polynomial time algorithm to test whether a given graph is outer $k$-planar, i.e., none of these recognition problems are NP-complete unless ETH fails. For the outer $k$-quasi-planar graphs we prove that, unlike other beyond-planar graph classes, every edge-maximal $n$-vertex outer $k$-quasi planar graph has the same number of edges, namely $2(k-1)n - \binom{2k-1}{2}$. We also construct planar 3-trees that are not outer $3$-quasi-planar. Finally, we restrict outer $k$-planar and outer $k$-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence on the boundary is a cycle in the graph. For each $k$, we express closed outer $k$-planarity and \emph{closed outer $k$-quasi-planarity} in extended monadic second-order logic. Thus, closed outer $k$-planarity is linear-time testable by Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Simultaneous Orthogonal Planarity

We introduce and study the $\textit{OrthoSEFE}-k$ problem: Given $k$ planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the $k$ graphs? We show that the problem is NP-complete for $k \geq 3$ even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for $k \geq 2$ even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for $k=2$ when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Planarity of Streamed Graphs

In this paper we introduce a notion of planarity for graphs that are presented in a streaming fashion. A $\textit{streamed graph}$ is a stream of edges $e_1,e_2,...,e_m$ on a vertex set $V$. A streamed graph is $\omega$-$\textit{stream planar}$ with respect to a positive integer window size $\omega$ if there exists a sequence of planar topological drawings $\Gamma_i$ of the graphs $G_i=(V,\{e_j \mid i\leq j < i+\omega\})$ such that the common graph $G^{i}_\cap=G_i\cap G_{i+1}$ is drawn the same in $\Gamma_i$ and in $\Gamma_{i+1}$, for $1\leq i < m-\omega$. The $\textit{Stream Planarity}$ Problem with window size $\omega$ asks whether a given streamed graph is $\omega$-stream planar. We also consider a generalization, where there is an additional $\textit{backbone graph}$ whose edges have to be present during each time step. These problems are related to several well-studied planarity problems. We show that the $\textit{Stream Planarity}$ Problem is NP-complete even when the window size is a constant and that the variant with a backbone graph is NP-complete for all $\omega \ge 2$. On the positive side, we provide $O(n+\omega{}m)$-time algorithms for (i) the case $\omega = 1$ and (ii) all values of $\omega$ provided the backbone graph consists of one $2$-connected component plus isolated vertices and no stream edge connects two isolated vertices. Our results improve on the Hanani-Tutte-style $O((nm)^3)$-time algorithm proposed by Schaefer [GD'14] for $\omega=1$.Comment: 21 pages, 9 figures, extended version of "Planarity of Streamed Graphs" (9th International Conference on Algorithms and Complexity, 2015