We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph.
The tester is given free access to a base graph $G=([\n],E)$, and oracle access to a function $f:E\to\{0,1\}$ that represents a subgraph of $G$.
The tester is required to distinguish between subgraphs that posses a predetermined property and subgraphs that are far from possessing this property.

We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model.
We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models.
Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds.

Correcting various minor errors and adding some clarifications.

We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph.
The tester is given free access to a base graph $G=([\n],E)$, and oracle access to a function $f:E\to\{0,1\}$ that represents a subgraph of $G$.
The tester is required to distinguish between subgraphs that posses a predetermined property and subgraphs that are far from possessing this property.

We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model.
We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models.
Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds.

Significantly revised Sec 3.2 and the second part of Sec 4.

We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph.
The tester is given free access to a base graph $G=([\n],E)$, and oracle access to a function $f:E\to\{0,1\}$ that represents a subgraph of $G$.
The tester is required to distinguish between subgraphs that posses a predetermined property and subgraphs that are far from possessing this property.

We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model.
We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models.
Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds.