Embedding in distance degree regular and distance degree injective graphs

Abstract

The eccentricity $e(u)$ of a vertex $u$ is the maximum distance of $u$ to any other vertex of $G$.The distance degree sequence (dds) of a vertex $u$ in a graph $G=(V, E)$ is a list of the number of vertices at distance $1,2, \ldots$, $e(u)$ in that order, where $e(u)$ denotes the eccentricity of $u$ in $G$. Thus the sequence $\left(d_{i_0}, d_{i_1}, d_{i_2}, \ldots, d_{i_j}, \ldots\right)$ is the dds of the vertex $v_i$ in $G$ where $d_{i_j}$ denotes number of vertices at distance $j$ from $v_i$. A graph is distance degree regular (DDR) graph if all vertices have the same dds. A graph is distance degree injective (DDI) graph if no two vertices have the same dds. In this paper, we consider the construction of a DDR graph having any given graph $G$ as its induced subgraph. Also we consider construction of some special class of DDI graphs.