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On the size and complexity of scrambles

Seamus Connor, Steven DiSilvio, Sasha Kononova, Ralph Morrison, Krish Singal

TL;DR

It is shown that there exist graphs with carton number exponential in the size of the graph, proving that scrambles are not valid NP certificates and finding that vertex congestion is an upper bound on screewidth and thus scramble number, leading to a new proof of the best known bound on the treewidth of line graphs.

Abstract

The scramble number of a graph, a natural generalization of bramble number, is an invariant recently developed to study chip-firing games and graph gonality. We introduce the carton number of a graph, defined to be the minimum size of a maximum order scramble, to study the computational complexity of scramble number. We show that there exist graphs with carton number exponential in the size of the graph, proving that scrambles are not valid NP certificates. We characterize families of graphs whose scramble number and gonality can be constant-factor approximated in polynomial time and show that the disjoint version of scramble number is fixed parameter tractable. Lastly, we find that vertex congestion is an upper bound on screewidth and thus scramble number, leading to a new proof of the best known bound on the treewidth of line graphs and a bound on the scramble number of planar graphs with bounded degree.

On the size and complexity of scrambles

TL;DR

It is shown that there exist graphs with carton number exponential in the size of the graph, proving that scrambles are not valid NP certificates and finding that vertex congestion is an upper bound on screewidth and thus scramble number, leading to a new proof of the best known bound on the treewidth of line graphs.

Abstract

The scramble number of a graph, a natural generalization of bramble number, is an invariant recently developed to study chip-firing games and graph gonality. We introduce the carton number of a graph, defined to be the minimum size of a maximum order scramble, to study the computational complexity of scramble number. We show that there exist graphs with carton number exponential in the size of the graph, proving that scrambles are not valid NP certificates. We characterize families of graphs whose scramble number and gonality can be constant-factor approximated in polynomial time and show that the disjoint version of scramble number is fixed parameter tractable. Lastly, we find that vertex congestion is an upper bound on screewidth and thus scramble number, leading to a new proof of the best known bound on the treewidth of line graphs and a bound on the scramble number of planar graphs with bounded degree.
Paper Structure (14 sections, 52 theorems, 34 equations, 6 figures, 1 table)

This paper contains 14 sections, 52 theorems, 34 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Treewidth and Scramble Number
  5. Carton Number
  6. Screewidth
  7. Premilinary results
  8. Properties of Carton Number
  9. Bounds on Carton Number
  10. Properties of disjoint scramble number
  11. Complexity of scramble number and variants
  12. Fixed parameter tractability of disjoint scramble number
  13. Approximability of scramble number and gonality
  14. Vertex Congestion and Screewidth

Key Result

Theorem 1.1

Let $G$ be a graph such that $\Delta(G) < \mathop{\mathrm{sn}}\nolimits(G)$. Then, $\mathop{\mathrm{cart}}\nolimits(G) \geq 3 \mathop{\mathrm{sn}}\nolimits(G) - |V(G)|$.

Figures (6)

  • Figure 1: Scrambles of orders $2$, $3$, and $1$, respectively. The second and third scrambles are disjoint, while the first is not.
  • Figure 2: Two scrambles of order three on a graph with carton number three.
  • Figure 3: A graph with two tree-cut decompositions of it, one of width $6$ and one of width $3$
  • Figure 4: A complete bipartite graph $K_{6, 4}$ with cycle added on the side with $6$ vertices.
  • Figure 5: A graph $G$ with a minor $G'$ of higher carton number; and a graph $H$ with an immersion minor $H'$ of higher carton number.
  • ...and 1 more figures

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 82 more