%A Wang,Guangfu
%A Li,Chenyang
%A Wang,Fengling
%D 2020
%J Frontiers in Physics
%C
%F
%G English
%K Hypercube,l 1 -embeddability,gate subgraph,gate-sum,Convex cut
%Q
%R 10.3389/fphy.2020.00146
%W
%L
%N 146
%M
%P
%7
%8 2020-June-04
%9 Original Research
%#
%! l 1 -Embeddability under Gate-sum Operation
%*
%<
%T l1-Embeddability Under Gate-Sum Operation of Two l1-Graphs
%U https://www.frontiersin.org/article/10.3389/fphy.2020.00146
%V 8
%0 JOURNAL ARTICLE
%@ 2296-424X
%X 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.