The geodesic cover problem for butterfly networks

TL;DR

This work resolves both vertex and edge geodesic cover numbers for the $r$-dimensional butterfly BF$(r)$. It combines structural analysis in two representations with a lower-bound via mapping to bipartite graphs and a constructive upper-bound using a three-stage diametral-cover, establishing ${\rm gcover}(BF(r)) = $ $\lceil (2/3) 2^{r} \rceil$ (for $r\ge5$; small cases checked) and ${\rm gcover_{e}}(BF(r)) = 2^{r}$ (for $r\ge3$). The results are underpinned by careful partitioning of level-0 and level-$r$ vertex sets, enumeration of maximal geodesics, and a cycle-based edge decomposition into isometric cycles. This advances theoretical understanding of geodesic covers in interconnection networks and sets the stage for similar analyses on other network topologies.

Abstract

A geodesic cover, also known as an isometric path cover, of a graph is a set of geodesics which cover the vertex set of the graph. An edge geodesic cover of a graph is a set of geodesics which cover the edge set of the graph. The geodesic (edge) cover number of a graph is the cardinality of a minimum (edge) geodesic cover. The (edge) geodesic cover problem of a graph is to find the (edge) geodesic cover number of the graph. Surprisingly, only partial solutions for these problems are available for most situations. In this paper we demonstrate that the geodesic cover number of the $r$-dimensional butterfly is $\lceil (2/3)2^r\rceil$ and that its edge geodesic cover number is $2^r$.

Figures (5)

• Figure 1: (a) Normal representation of ${\rm BF}(3)$ (b) Diamond representation of ${\rm BF}(3)$.
• Figure 2: (a) Vertices from $U^{b}$ and $W^{b}$ are blue, vertices from $U^{r}$ and $W^{r}$ are red. (b) $v_{j}$ is an internal vertex of $P$ and of degree $2$. Its neighbors $v_{j-1}$ and $v_{j+1}$ lie in $P$. One of the adjacent vertices $\{v_{j-1}$, $v_{j+1}\}$ lie in $G_1$ and the other lie in $G_2$.
• Figure 3: (a) The complete bipartite graph $G'$. (b) The complete bipartite graph $G"$.
• Figure 5: In Stage $1$, geodesics $P_{u_{1}}(w_{1},w_{9})$, $P_{u_{2}}(w_{2},w_{10})$ and $P_{u_{16}}(w_{16},w_{8})$, $P_{u_{15}}(w_{15},w_{7})$ are constructed (pink). In Stage $2$, geodesics $P_{w_{3}}(u_{3},u_{4})$, $P_{w_{5}}(u_{5},u_{6})$ and $P_{w_{14}}(u_{14},u_{13})$, $P_{w_{12}}(u_{12},u_{11})$ are constructed (green). The vertices which are not covered in previous stages are circled. They are then covered in Stage $3$.
• Figure 6: Here $U = \{u^{r}_{1}, u^{r}_{2}, u^{r}_{3}, u^{b}_{1}, u^{b}_{2}, u^{b}_{3}\}$ and $W = \{w^{r}_{1}, w^{r}_{2}, w^{r}_{3}, w^{b}_{1}, w^{b}_{2}, w^{b}_{3}\}$, where $u^{r}_{1}, u^{r}_{2}, u^{r}_{3}, w^{r}_{1}, w^{r}_{2}, w^{r}_{3}$ are red vertices, and $u^{b}_{1}, u^{b}_{2}, u^{b}_{3}, w^{b}_{1}, w^{b}_{2}, w^{b}_{3}$ are blue vertices. How to cover the vertices by geodesics of length $3$ with end vertices in opposite colors?

Theorems & Definitions (13)

• Lemma 2.1
• Proposition 2.2
• Lemma 2.3
• Corollary 2.4
• Lemma 2.5
• Corollary 2.6
• Lemma 3.1
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5