One of the pursuit-evasion games is the game of cops and a robber, which is 
played on graph G=(V,E). Aigner and Fromme proved that, if we have the 
shortest path P between two vertices in G, then one cop guarantees that the 
robber will be caught, if he moves onto P. Very similar result is from 2001, 
when Fitzpartick and Nowakowski proved that one cop will catch a robber, if 
he steps on some isometric path P' in G. In the contents of the game of 
guarding a subgraph, P and P' can be guarded by one guard. In this seminar, we
generalize the concept of guarding to arbitrary subgraphs of a graph. We introduce a
new concept, the shadow of the intruder, which maps the location of the intruder to
the set of vertices of G, from which a guard can prevent the intruder from entering
the guarded subgraph. We describe the characteristic properties of this function and
prove that a single guard can guard the subgraph if and only if he is in the shadow
of the intruder.

IMFM and Univerza v Ljubljani, 
Fakulteta za matematiko in fiziko
Jadranska 21 (the new building)