  We consider the constrained graph alignment problem which has applications in
biological network analysis. Given two input graphs $G_1=(V_1,E_1),
G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if
the vertex pairs are adjacent in their respective graphs. %In general terms The
goal is to provide a one-to-one mapping between the vertices of the input
graphs in order to maximize edge conservation. However the allowed mappings are
restricted since each vertex from $V_1$ (resp. $V_2$) is allowed to be mapped
to at most $m_1$ (resp. $m_2$) specified vertices in $V_2$ (resp. $V_1$). Most
of results in this paper deal with the case $m_2=1$ which attracted most
attention in the related literature. We formulate the problem as a maximum
independent set problem in a related {\em conflict graph} and investigate
structural properties of this graph in terms of forbidden subgraphs. We are
interested, in particular, in excluding certain wheals, fans, cliques or claws
(all terms are defined in the paper), which corresponds in excluding certain
cycles, paths, cliques or independent sets in the neighborhood of each vertex.
Then, we investigate algorithmic consequences of some of these properties,
which illustrates the potential of this approach and raises new horizons for
further works. In particular this approach allows us to reinterpret a known
polynomial case in terms of conflict graph and to improve known approximation
and fixed-parameter tractability results through efficiently solving the
maximum independent set problem in conflict graphs. Some of our new
approximation results involve approximation ratios that are function of the
optimal value, in particular its square root; this kind of results cannot be
achieved for maximum independent set in general graphs.

Data Structures and Algorithms;Computational Engineering, Finance, and Science