## Section: New Results

### Subgraph Epimorphisms

Participants : François Fages, Steven Gay, Thierry Martinez, Francesco Santini, Sylvain Soliman.

The operations of deleting and merging vertices are natural operations for reducing a graph. While graph reductions through a sequence of vertex deletions (resp. mergings) characterize subgraph isomorphisms (resp. graph epimorphisms), sequences of both vertex deletion and merging operations characterize subgraph epimorphisms. Our proposal is thus to use subgraph epimorphism for comparing graphs in applications in systems biology and image analysis, when a more flexible notion than the classical notion of subgraph isomorphism is required.

In collaboration with Christine Solnon (INSA Lyon), we have developed the theory of subgraph epimorphisms. We have defined the SEPI, EPI and SISO distances between two graphs as the size of the largest SEPI (resp. EPI, SISO) lower bound graphs. These distances are equal to the minimum number of respectively vertex deletion and/or merging operations that are necessary to obtain isomorphic graphs. They are also metrics on graphs and we have ${d}_{d}\ge {d}_{md}$ and ${d}_{m}\ge {d}_{md}$. From a computational point of view, we have shown that the existence of a SEPI between two graphs is an NP-complete problem and have presented a constraint satisfaction algorithm for solving it.

Our algorithm is implemented in BIOCHAM and is currently improved for better performance on large graphs and generalized as a SEPI graph constraint propagation algorithm for computing SEPI lower and upper bounds.