Word-Representation of Melon Graphs

TL;DR

Let M be a melon graph, i.e., a theta-like graph formed by $m$ vertex-disjoint $E_i$ between two endpoints. The paper proves that $\mathcal{R}(M)\le 3$ (melon graphs are 3-word-representable) and provides a dichotomy: $\mathcal{R}(M)=2$ when all $E_i$ have length at most two or share parity, and $\mathcal{R}(M)=3$ otherwise, with subfamilies such as $M_3$ and $B_3$ determining the boundary. For comparability melon graphs $M_c$, it shows $\mathcal{R}^p(M_c)\le 3$ with explicit 3-permutation constructions, and characterizes comparability via parity and endpoint adjacency; it further analyzes line graphs $L(M)$, showing $L(M)$ is word-representable iff $M$ avoids $A_3$, and that $\mathcal{R}(L(M))\le 3$, achieving 3 precisely when $M$ has three long paths (i.e., $L(M)$ contains $K_3\square K_2$ as a vertex-minor). These results tie melon graphs to planar posets and motivate extensions to series-parallel graphs and forbidden-subgraph characterizations for line graphs in this class.

Abstract

The notion of word-representable graphs is a generalization of comparability graphs, in which graphs are represented by words. The complexity of word-representation of a word-representable graph is captured through the representation number, whereas the corresponding concept is the permutation-representation number for comparability graphs. The graphs with the (permutation-)representation number at most two were characterized in the literature. While certain examples in the class of graphs with the (permutation-)representation number three are known, no characterization for these classes is available. In this work, we prove that the representation number of melon graphs is at most three. Further, we characterize the class of melon graphs restricted to comparability graphs and show that their permutation-representation number is also at most three. Moreover, this work characterizes the word-representable line graphs of melon graphs and establishes that their representation number is at most three.

Figures (16)

• Figure 1: $M_3$
• Figure 2: Local complementation of $M_3$ at 0'
• Figure 3: $B_3$
• Figure 4: Local complementation of $B_3$ at 0'
• Figure 5: Melon graph $M_c$

Theorems & Definitions (47)

• theorem 1: kitaev08
• theorem 2
• proof
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• lemma 4