Section: New Results

Theory of Subgraph Epimorphisms

Participants : François Fages, Steven Gay, Thierry Martinez, 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 where 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 in [5] . We have shown that SEPIs preserve graph completeness and arc symmetry and that, just like SISO and EPI, SEPI is not a well quasi order. 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. This algorithm is implemented in BIOCHAM .

It is worth noticing that, given two graphs $G$ and $G^{\text{'}}$, the greatest lower SEPI bounds and the least upper SEPI bounds are also interesting to compute since they represent “intersection” and “union” graphs for the SEPI relation. For instance, in our motivating application in systems biology, these objects correspond to the intersection (resp. union) of models at different levels of details for a given biochemical process. These graphs are not unique but we are confident that the constraint satisfaction algorithm presented in [10] can be interestingly generalized to compute them.