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Spanning tree congestion of proper interval graphs

Yota Otachi

Abstract

We show that the spanning tree congestion problem is NP-complete even for proper interval graphs of linear clique-width at most 4.

Spanning tree congestion of proper interval graphs

Abstract

We show that the spanning tree congestion problem is NP-complete even for proper interval graphs of linear clique-width at most 4.
Paper Structure (15 sections, 6 theorems, 8 equations, 2 figures)

This paper contains 15 sections, 6 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related results
  3. On a recently announced algorithm for interval graphs
  4. Preliminaries
  5. An auxiliary lemma
  6. Proof of \ref{['thm:stc-pint']}
  7. Construction
  8. Graph class
  9. Equivalence
  10. Vertex degrees
  11. Concluding remarks
  12. Graph classes
  13. Fixed $k$
  14. Approximation
  15. Structural parameterizations

Key Result

Theorem 1

Spanning Tree Congestion is $\mathrm{NP}$-complete on proper interval graphs of linear clique-width at most $4$.

Figures (2)

  • Figure 1: The complexity of STC on graph classes. The (red) solid rectangles represent NP-complete cases and the (green) rounded rectangles represent polynomial-time solvable cases.
  • Figure 2: The neighborhoods of $y_{p}, y_{q}, y_{r}$ in $X$ and $Z$, where $p \in [m]$ and $q, r \in [m+1, a_{3m}]$.

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 9 more