2-switch: transition and stability on forests and pseudofests

Abstract

Given any two forests (pseudoforests) with the same degree sequence, we show that one can be transformed into the other by a sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests (pseudoforests). We also prove that the 2-switch operation perturbs minimally some well-known integer parameters in families of graphs with the same degree sequence. Then, we apply these results to conclude that the studied parameters have the interval property on those families.

Figures (3)

• Figure 1: Every 2-switch sequence transforming $B_3$ into $B'_3$ must pass through a non-bipartite graph.
• Figure 2: Every 2-switch sequence transforming $N_4$ into $N'_4$ must pass through a bipartite graph.
• Figure 3: $\delta$ is not stable in $\mathcal{F}(d)$: $T'={{a \ b}\choose{c \ d}}T$, but $\delta(T')-\delta(T)=2$

Theorems & Definitions (85)

• Theorem 1.1
• Theorem 1.2: Taylor
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3.1
• proof
• Lemma 3
• proof