We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries.
We also explain how the proof can be adapted to show that any hereditary property of $2$-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable.
The first result generalizes the result of Alon and Shapira [FOCS'05; SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable.
The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties.
The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family $\mathcal{F}$ of forbidden vertex-ordered graphs, and any $\epsilon > 0$, there exist $\delta > 0$ and $k$ so that any vertex-ordered graph which is $\epsilon$-far from being $\mathcal{F}$-free contains at least $\delta n^{|F|}$ copies of some $F\in\mathcal{F}$ (with the correct vertex order) where $|F|\leq k$.
The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for $k$-partite graphs with ``undesirable'' edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph.

Updated abstract

We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries.
We also explain how the proof can be adapted to show that any hereditary property of $2$-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable.
The first result generalizes the result of Alon and Shapira [FOCS'05; SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable.
The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties.
%To the best of our knowledge, these are the first testing results of this kind for the ordered versions of graphs and matrices.

The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family $\mathcal{F}$ of forbidden vertex-ordered graphs, and any $\epsilon > 0$, there exist $\delta > 0$ and $k$ so that any vertex-ordered graph which is $\epsilon$-far from being $\mathcal{F}$-free contains at least $\delta n^{|F|}$ copies of some $F\in\mathcal{F}$ (with the correct vertex order) where $|F|\leq k$.
The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for $k$-partite graphs with ``undesirable'' edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph.