l1-Embeddability Under Gate-Sum Operation of Two l1-Graphs - Wang, Guangfu; Li, Chenyang; Wang, Fengling (Frontiers in Physics, 2020)
An l1-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 l1-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 G1 and G2 with respect to H if H is a gate subgraph of at least one of G1 and G2, such that G1∪G2 = G, G1∩G2 = H, and both G1 and G2 are isometric subgraphs of G. In this article, we have shown that the gate-sum graph of two l1-graphs is also an l1-graph.