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Local 2-separators

Johannes Carmesin

Abstract

How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing 'Local Separators' of graphs. Applications include: 1. A unique decomposition theorem for graphs along their local 2-separators analogous to the 2-separator theorem; 2. an exact characterisation of graphs with no bounded subdivision of a wheel.

Local 2-separators

Abstract

How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing 'Local Separators' of graphs. Applications include: 1. A unique decomposition theorem for graphs along their local 2-separators analogous to the 2-separator theorem; 2. an exact characterisation of graphs with no bounded subdivision of a wheel.

Paper Structure

This paper contains 11 sections, 27 theorems, 3 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Constructive perspective
  3. Explorer neighbourhood
  4. Intermezzo: Block-Cutvertex Graphs
  5. The existential statement of the local 2-separator theorem
  6. Properties of local 2-separators
  7. When all local 2-separators are crossed...
  8. The uniqueness statement of the local 2-separator theorem
  9. Graph-Decompositions
  10. Outlook
  11. Appendix I: an alternative proof for the existential statement

Key Result

Theorem 1.1

For every $r\in {\mathbb{N}}\cup\{\infty\}$, every connected $r$-locally 2-connected graph $G$ has a graph-decomposition of adhesion two and locality $r$ such that all its torsos are $r$-locally 3-connected or cycles of length at most $r$. Moreover, the separators of this graph-decomposition are the

Figures (9)

  • Figure 1: The graph $C_6 \boxtimes K_1$.
  • Figure 2: The balls $B_{r/2}(v)$ and $B_{r/2}(w)$ are marked by grey stripes, in rising and falling patters, respectively. Two paths between the vertices $x$ and $y$, one from either ball, form a cycle that is contained in neither ball.
  • Figure 3: On the right we depicted the explorer-neighbourhood $\mathop{\mathrm{Ex_r}}\nolimits(v,w)$ of the graph on the left. The value for $r/2$ is seven. Here the grey paths all have length equal to $(r/2)-2$. The core is just the path of length four between $v$ and $w$. The cycle of length $r$ is still a cycle in $\mathop{\mathrm{Ex_r}}\nolimits(v,w)$ since as a cycle it is included in both $B_{r/2}(v)$ and $B_{r/2}(w)$, see \ref{['unique_copy_extended']} for details. The cycle of length $r+2$ is not contained in one of the balls $B_{r/2}(v)$ or $B_{r/2}(w)$ and hence some of its vertices get two copies in $\mathop{\mathrm{Ex_r}}\nolimits(v,w)$. Indeed, the vertex $x$ has distance at most $r$ from both vertices $v$ and $w$. Still it has the two copies $x_1$ and $x_2$ in the explorer-neighbourhood.
  • Figure 4: The graph on the left is obtained from the graph on the right by locally splitting at the local 2-separator given by the blue edge. This blue edge gets two copies on the left, one for each local component. In \ref{['sec:graph-deco']} we shall investigate the inverse operation of local cutting.
  • Figure 5: An alternating cycle. The vertices $a_1$ and $a_2$ of the first local 2-separator are indicated by boxes, the vertices $b_1$ and $b_2$ of the second local 2-separator are indicated by crosses. The cyclic order of the cycle induced on these four vertices alternates between the two local separators.
  • ...and 4 more figures

Theorems & Definitions (118)

  • Example 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 1
  • Definition 1
  • Lemma 1
  • proof
  • Remark 2
  • Definition 2: Local 2-sum
  • ...and 108 more