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A note on distance-hereditary graphs whose complement is also distance-hereditary

Hugo Jacob

Abstract

Distance-hereditary graphs are known to be the graphs that are totally decomposable for the split decomposition. We characterise distance-hereditary graphs whose complement is also distance-hereditary by their split decomposition and by their modular decomposition.

A note on distance-hereditary graphs whose complement is also distance-hereditary

Abstract

Distance-hereditary graphs are known to be the graphs that are totally decomposable for the split decomposition. We characterise distance-hereditary graphs whose complement is also distance-hereditary by their split decomposition and by their modular decomposition.

Paper Structure

This paper contains 13 theorems, 4 figures.

Key Result

Lemma 1

Two vertices $u,v \in V(G)$ are adjacent in $G$ if and only if, for every internal node $t$ of $T$ on the path $P$ between the leaves mapped to $u$ and $v$, there is an edge in $\mu(t)$ between the vertices of $\mu(t)$ mapped to the edges on $P$.

Figures (4)

  • Figure 1: The forbidden induced subgraphs corresponding to the class of distance-hereditary. From left to right: holes (cycles of length at least $5$), house, domino, and gem.
  • Figure 2: The forbidden induced subgraphs of DH $\cap$ co-DH. From left to right: $C_5$, the House and its complement $P_5$, the Gem and its complement $K_1 + P_4$.
  • Figure 3: The only two prime graphs in DH $\cap$ co-DH and their split decomposition. The graphs $\mu(t)$ are depicted inside light blue circles and edges of $T$ are outside these circles. From left to right: $P_4$ and the Bull.
  • Figure 4: Two configurations in the proof of \ref{['lem:cutvertices']}.

Theorems & Definitions (18)

  • Lemma 1: GioanP12
  • Theorem 1: GioanP12
  • Theorem 2: GioanP12
  • Theorem 3: twwOne
  • Theorem 4
  • Theorem 5
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 8 more