Check if two graphs are edge-disjoint

Question

Two undirected graphs $G$ and $H$ on the vertices $1,2,\ldots,n$ are disjoint if the intersection of their edge sets is empty. Assume both $G$ and $H$ are represented by adjacency matrices.

Describe an efficient algorithm that decides if $G$ and $H$ are disjoint. What is the complexity of your algorithm? Justify the correctness of your algorithm and your complexity claim.

What I am thinking is just picking an arbitrary vertex on either graph like $G$ and checking if we can reach a vertex from the other graph $H$ by running some simple traversal like DFS. Is there a better way? I am thinking what if we go through the adjacency matrix of one graph and check if any edge connects a vertex from one graph to the other.

Answer 1

The adjacency matrix $A_G$ of an undirected graph $G=(V,E)$ is defined as follows: $A_G$ is a $V \times V$ matrix, $A_G(v,v) = 0$ for all $v \in V$, and $A_G(u,v) = 1$ if $\{u,v\} \in E$ and $A_G(u,v) = 0$ if $\{u,v\} \notin E$.

Two graphs $G,H$ are edge-disjoint if there doesn't exist an edge $\{u,v\}$ which belongs to both of them. That is, $G,H$ are edges-disjoint if there do no exist $u,v$ such that $A_G(u,v) = A_H(u,v) = 1$.

You take it from here.