AUTHOR=Wang Guangfu, Li Chenyang, Wang Fengling
TITLE=l1-Embeddability Under Gate-Sum Operation of Two l1-Graphs
JOURNAL=Frontiers in Physics
VOLUME=8
YEAR=2020
PAGES=146
URL=https://www.frontiersin.org/article/10.3389/fphy.2020.00146
DOI=10.3389/fphy.2020.00146
ISSN=2296-424X
ABSTRACT=An l_{1}-graph is one in which the vertices can be labeled by binary vectors such that the Hamming distance between two binary addresses is, to scale, the distance in the graph between the corresponding vertices. This study was designed to determine whether the gate-sum operation can inherit the l_{1}-embeddability. The subgraph H of a graph G is called a gate subgraph if, for every vertex v ∈ V(G), there exists a vertex x ∈ V(H) such that for every vertex u of H, x lies on a shortest path from v to u. The graph G is defined as the gate-sum of two graphs G_{1} and G_{2} with respect to H if H is a gate subgraph of at least one of G_{1} and G_{2}, such that G_{1}∪G_{2} = G, G_{1}∩G_{2} = H, and both G_{1} and G_{2} are isometric subgraphs of G. In this article, we have shown that the gate-sum graph of two l_{1}-graphs is also an l_{1}-graph.