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Beyond Outerplanarity

Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre Löffler, Alexander Wolff

TL;DR

This work studies convex drawings of graphs restricted to outer $k$-planar and outer $k$-quasi-planar classes, deriving structural and algorithmic consequences. It proves a tight-looking degeneracy bound of $\big\lfloor 3.5\sqrt{k}\rfloor$, a balanced separator bound of $2k+3$, and quasi-polynomial recognition for fixed $k$, with ETH-based implications. It further shows incomparability between outer $k$-quasi-planar and planar graphs and provides linear-time testing for full/closed variants via MSO$_2$ encodings and Courcelle’s theorem, leveraging a treewidth bound of $3k+11$. The results unify combinatorial structure with algorithmic tractability and raise open questions about tighten bounds and complexity of related recognition problems.

Abstract

We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer $k$-planar} graphs, where each edge is crossed by at most $k$ other edges; and, \emph{outer $k$-quasi-planar} graphs where no $k$ edges can mutually cross. We show that the outer $k$-planar graphs are $\lfloor3.5\sqrt{k}\rfloor$-degenerate, and consequently that every outer $k$-planar graph can be colored with $\lfloor3.5\sqrt{k}\rfloor + 1$ colors. We further show that every outer $k$-planar graph has a balanced vertex separator of size at most $2k+3$. For each fixed $k$, these small balanced separators allow us to test outer $k$-planarity in quasi-polynomial time, e.g., this implies that none of these recognition problems is NP-hard unless the Exponential Time Hypothesis fails. We also show that the class of outer $k$-quasi-planar graphs and the class of planar graphs are incomparable. Finally, we restrict outer $k$-planar and outer $k$-quasi-planar drawings to \emph{full} drawings (where no crossing appears on the boundary of the outer face) and to \emph{closed} drawings (where the vertex sequence on the boundary of the outer face is a Hamiltonian cycle in the graph). For each $k$, we express \emph{closed outer $k$-planarity} and \emph{closed outer $k$-quasi-planarity} in \emph{extended monadic second-order logic}. Due to a result of Wood and Telle (New York J. Math., 2007) every outer $k$-planar graph has treewidth at most $3k+11$. Thus, Courcelle's theorem implies that closed outer $k$-planarity is linear time testable. We leverage this result to further show that full outer $k$-planarity can also be tested in linear time.

Beyond Outerplanarity

TL;DR

This work studies convex drawings of graphs restricted to outer -planar and outer -quasi-planar classes, deriving structural and algorithmic consequences. It proves a tight-looking degeneracy bound of , a balanced separator bound of , and quasi-polynomial recognition for fixed , with ETH-based implications. It further shows incomparability between outer -quasi-planar and planar graphs and provides linear-time testing for full/closed variants via MSO encodings and Courcelle’s theorem, leveraging a treewidth bound of . The results unify combinatorial structure with algorithmic tractability and raise open questions about tighten bounds and complexity of related recognition problems.

Abstract

We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer -planar} graphs, where each edge is crossed by at most other edges; and, \emph{outer -quasi-planar} graphs where no edges can mutually cross. We show that the outer -planar graphs are -degenerate, and consequently that every outer -planar graph can be colored with colors. We further show that every outer -planar graph has a balanced vertex separator of size at most . For each fixed , these small balanced separators allow us to test outer -planarity in quasi-polynomial time, e.g., this implies that none of these recognition problems is NP-hard unless the Exponential Time Hypothesis fails. We also show that the class of outer -quasi-planar graphs and the class of planar graphs are incomparable. Finally, we restrict outer -planar and outer -quasi-planar drawings to \emph{full} drawings (where no crossing appears on the boundary of the outer face) and to \emph{closed} drawings (where the vertex sequence on the boundary of the outer face is a Hamiltonian cycle in the graph). For each , we express \emph{closed outer -planarity} and \emph{closed outer -quasi-planarity} in \emph{extended monadic second-order logic}. Due to a result of Wood and Telle (New York J. Math., 2007) every outer -planar graph has treewidth at most . Thus, Courcelle's theorem implies that closed outer -planarity is linear time testable. We leverage this result to further show that full outer -planarity can also be tested in linear time.

Paper Structure

This paper contains 13 sections, 11 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Concepts
  4. Contribution
  5. Outer k-Planar Graphs
  6. Degeneracy
  7. Quasi-Polynomial-Time Recognition via Balanced Separators
  8. Outer $k$-Quasi-Planar Graphs
  9. Testing for Full Convex Drawings via MSO$_2$
  10. Discussions and Open Problems
  11. SAT Formulations
  12. Outer Quasi-Planarity Checker
  13. Page Number Checker

Key Result

Theorem 2

For every positive integer $k$, let $\delta^\star$ be the largest minimum degree among all outer $k$-planar graphs. Then ${\delta^\star \le \lfloor c_k\sqrt{k} \rfloor }$, where The sequence $(c_k)_{k\ge1}$ is monotonically decreasing with $c_1 = 3.5$ and limit $3\sqrt{2}/2$.

Figures (3)

  • Figure 2: Shapes of separators, special separator $S$ in blue, regions in different colors (red, orange, and pink), components connected to blue vertices in green: (a) closest-parallels case; (b) single-edge case; (c) special case for single-edge separators.
  • Figure 3: Outer quasi-planar drawings of the graphs $K_5$ and $K_{4,4}$
  • Figure 4: A planar 3-tree (with 23 vertices) that is not outer quasi-planar: (a) planar drawing; (b) deleting the blue vertex makes the drawing outer quasi-planar.

Theorems & Definitions (12)

  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Proposition 7
  • Proposition 8
  • Theorem 9
  • Remark 10
  • Theorem 12: courcelle1990courcelle2012
  • ...and 2 more