Contractions in perfect graph

TL;DR

This work tackles contraction perfection: graphs that remain perfect after contracting any edge set. Using the strong perfect graph theorem, it shows contraction perfection is equivalent to preserving perfection under any single-edge contraction, and then derives a forbidden-subgraph characterization: no hole of size $\ge 5$, no odd antihole, and no expanded antihole. A central construction, the utter graph $u(G)$, links contraction perfection to graph perfection by establishing a bijection between co-2-plexes of $G$ and stable sets of $u(G)$; consequently, $G$ is contraction perfect iff $u(G)$ is perfect, yielding a polynomial-time method to compute maximum weight co-2-plex via stable-set optimization in perfect graphs. The paper further explores implications for recognition and for several graph classes (split, trivially perfect, interval, chordal), and discusses the replication of contraction-perfectness under vertex twins. Overall, it provides a cohesive framework that connects edge contractions, forbidden minor-like characterizations, and the tractability of co-2-plex optimization in contraction-perfect graphs.

Abstract

In this paper, we characterize the class of {\em contraction perfect} graphs which are the graphs that remain perfect after the contraction of any edge set. We prove that a graph is contraction perfect if and only if it is perfect and the contraction of any single edge preserves its perfection. This yields a characterization of contraction perfect graphs in terms of forbidden induced subgraphs, and a polynomial algorithm to recognize them. We also define the utter graph $u(G)$ which is the graph whose stable sets are in bijection with the co-2-plexes of $G$, and prove that $u(G)$ is perfect if and only if $G$ is contraction perfect.

Figures (4)

• Figure 1: An expanded antihole.
• Figure 2: Illustration of the proof of Lemma \ref{['lem:expandedAntiholes']} (the doted lines represent nonedges and dots the remaining vertices of the antipath).
• Figure 3: Illustration for the proof of Theorem \ref{['the:contractequiv']}: the odd antihole $H_u \cup u' \setminus \{w_i, w_{i+1}\}$ of $G'$ is given in light gray
• Figure 4: A graph $G$ and its utter graph $u(G)$.

Theorems & Definitions (16)

• Lemma 1.1
• Lemma 1.2
• Definition 1.3
• Lemma 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Lemma 1.8
• Corollary 1.10
• Lemma 2.1