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Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

Glencora Borradaile, Hung Le, Melissa Sherman-Bennett

Abstract

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices, assuming that the connected components of the inner layer are two-connected.

Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

Abstract

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices, assuming that the connected components of the inner layer are two-connected.

Paper Structure

This paper contains 13 sections, 6 theorems, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. Large induced forests of planar graphs: known results
  3. Outline of our results
  4. Definitions
  5. Block-Cut Tree.
  6. Planar graphs.
  7. Planar duality.
  8. Outerplanarity.
  9. Facial Block.
  10. 2-outerplanar graphs have vertex-arboricity 2
  11. 2-outerplanar graphs have large induced outerplanar graphs
  12. Future directions
  13. Acknowledgments:

Key Result

Theorem 1

If $G$ is a 2-outerplanar graph, then the vertex arboricity of $G$ is at most $2$: $a(G) \leq 2$.

Figures (6)

  • Figure 1: The critical triangle $abc$ and two neighbors $d,e$ of $c$ in $L_1$. Hollow vertices are in $L_2$.
  • Figure 2: The critical triangle $abc$ with edge $fa \in \partial G[L_2]$. Hollow vertices are in $L_2$.
  • Figure 3: A pair of critical triangles $abc$ and $bfg$. Hollow vertices are in $L_2$
  • Figure 4: The outerplanar induced subgraph of this graph found by the algorithm in Theorem \ref{['thm:2outer']} is induced by the white vertices. Every induced forest on at least half of the vertices of this graph (an example is shown by the bolded edges) includes vertices not in this outerplanar subgraph.
  • Figure 5: A 2-outerplanar graph whose largest induced outerplanar subgraph is on $\frac{2}{3}$ of its vertices. The white vertices induce such a subgraph, found by the algorithm in Theorem \ref{['thm:2outer']}.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Theorem 1
  • proof
  • Claim 4
  • proof
  • Claim 5
  • proof
  • Claim 6
  • proof
  • Claim 8
  • proof
  • ...and 19 more