Planarity of Mycielski-like graph expansions

TL;DR

This paper characterizes when the great shadow $S(G)$ of a graph is planar, proving a sharp equivalence: $S(G)$ is planar if and only if $G$ is a bipartite cactus. The authors show the necessity of this condition by identifying obstructions in $S(G)$—topological $K_{3,3}$ arising from odd cycles and from non-cactus structures via a $K_4^-$—and establish sufficiency through a recursive, cycle-tree based construction that embeds $S(G)$ for bipartite cacti. The base case handles even cycles with an explicit planar embedding, and the inductive step glues subdrawings along shared edges to preserve planarity. The results connect a Mycielski-like graph expansion to practical circuit-routing considerations on single-layer boards, providing a precise structural criterion for planarity in this graph-analytic setting.

Abstract

For a graph $G$, we define its great shadow $S(G)$ as a construction that duplicates each vertex $v$ in $G$ and sets this duplicated vertex adjacent to $v$ and all neighbors of $v$. Great graph shadows arise naturally in the routing of diode-and-switch circuits for computer keyboards, and are closely related to the Mycielski operation. These diode-and-switch circuits can be routed on a single-sided printed-circuit board if and only if the corresponding great shadow is planar. In this paper, we characterize all graphs with planar great shadows. Such graphs are always bipartite cactus graphs.

Figures (7)

• Figure 1: The graph $K_3$ is on the left, and its great shadow, $S(K_3)$, is on the right. A $K_{3,3}$ subgraph witnessing the non-planarity of $S(K_3)$ is highlighted in blue.
• Figure 2: The graph $C_4 + v$ is on the left, and a planar drawing of its great shadow is on the right.
• Figure 3: The even case (Case 1) of Lemma \ref{['lem:cactusnecc']}. The figure pictures $S(\theta (4, 6, 4))$. Non-gray edges highlight the embedded $T K_{3,3}$. White vertices are shadow vertices, and black vertices are originals. Blue vertices ($a'_2$, $a'_1$ and $u'$) and red vertices ($a_1$, $a_2$, $v$) form partitions $\Delta_2$ and $\Delta_1$ respectively. Black edges are the isolated edges $u'v$, $u'a_2$, $a'_1v$ and $a'_2v$. Colored edges lie on paths. The green path is $P_{a'_2v}$, the cyan path is $P_{u'a_2}$, the orange path is $P_{u'v}$, and the violet path is $P_{a'_1v}$.
• Figure 4: The odd case (Case 2) of Lemma \ref{['lem:cactusnecc']}. The figure pictures $S(\theta (5, 7, 3))$. As in Figure \ref{['fig:evencase']}, the non-gray edges highlight the embedded $T K_{3,3}$. White vertices are shadow vertices, and black vertices are originals. Blue vertices (now $a'_1$, $a'_2$, and $v$) and red vertices (now $a_1$, $a_2$, $u$) form partitions $\Delta_2$ and $\Delta_1$ respectively. Black edges are the isolated edges $a_1 a'_1$, $a_2 a'_2$, $a_2 a'_1$, $a_1 a'_2$ and $a'_1 u$. Colored edges lie on paths. The green path is $P_{a_2v}$, the cyan path is $P_{a'_2u}$, the orange path is $P_{u v}$, and the violet path is $P_{a_1 v}$.
• Figure 5: The cycle graph $C_{16}$ is on the left. On the right is a a planar drawing of $S(C_{16})$, produced by the method described within the proof of Lemma \ref{['lem:evenplanar']}.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2: Garza and Shinkel garza1999shadow
• Definition 3: Great shadow graph, $S(G)$
• Lemma 4
• proof
• Theorem 5: Kuratowski kuratowski1930probleme
• Lemma 6
• Lemma 7
• Definition 8
• Definition 9: Cycle tree