A graph reconstruction problem involving common neighbors
Given a simple graph $G = (V, E)$ on $v$ vertices and two distinct vertices $x, y \in V$, the co-degree $c_{x,y}$ associated to the pair $\{x, y\}$ is the number of their common neighbors in the graph $G$. The co-degree sequence of $G$, denoted by $\gamma(G)$, is the list of all the co-degrees associated to all the possible pairs of distinct vertices, arranged in non-increasing order. In this paper we consider the following problem, which can be viewed as a generalization of a result by Erd\H{o}s and Gallai as well as of the Erd\H{o}s, R\'enyi and S\'os' friendship theorem: given an integer $v\geqslant 2$ and a sequence $\gamma$ of nonnegative integers arranged in non-increasing order, establish if there exists a simple graph on $v$ vertices having $\gamma$ as its co-degree sequence and, in case of positive answer, provide such a graph. We provide a full answer to this problem for the class of planar $C_4$-free graphs.