Maximal planar graphs that embed as centers

TL;DR

The paper addresses which maximal planar graphs can appear as the center of a planar graph. It introduces quasi-eccentricity and the quasi-eccentric face criterion, proving a general necessary condition and, for maximal planar graphs, a constructive sufficiency via face-based gadget attachments that realize a target eccentricity. It then shows that every maximal planar graph with order at most 8 can indeed be realized as a center (with a sharp counterexample at order 9), yielding a precise boundary for the result. The work provides exact characterizations, practical embedding constructions, and invites further questions on broader graph classes and center connectivity.

Abstract

A maximal planar graph is a graph which can be embedded in the plane such that every face of the graph is a triangle. The center of a graph is the subgraph induced by the vertices of minimum eccentricity. We introduce the notion of quasi-eccentric vertices, and use this to characterize maximal planar graphs that are the center of some planar graph. We also present some easier to check only necessary / only sufficient conditions for planar and maximal planar graphs to be the center of a planar graph. Finally, we use the aforementioned characterization to prove that all maximal planar graphs of order at most 8 are the center of some planar graph -- and this bound is sharp.

Figures (18)

• Figure 1: Given any graph $G$, the Hedetniemi construction yields a graph $H(G)$ with $G$ as its center. In the example above, the vertices and edges of $G$ are bold.
• Figure 2: The seven possible centers of a maximal outerplanar graph.
• Figure 3: A maximal planar graph with center $2K_3$. The central vertices are black.
• Figure 4: The path graph $G : v_1, v_2, v_3, v_4, v_5$. The vertices of the set $S = \{v_2, v_3\}$ are coloured grey.
• Figure 5: The bold cycle on the left is a separating cycle which is not a Jordan separating cycle. The bold cycle on the right is a Jordan separating cycle.

Theorems & Definitions (59)

• Lemma 1
• proof
• Theorem 3: The quasi-eccentric face criterion
• proof
• Corollary 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7