l1-Embeddability Under Gate-Sum Operation of Two l1-Graphs - Wang, Guangfu; Li, Chenyang; Wang, Fengling (Frontiers in Physics, 2020)
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, up to scale, the distance in the graph of corresponding vertices. This study is 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 show that the gate-sum graph of two l 1 -graphs is also an l 1 -graph.