Abstract: The graph-searching problem is a well-known $NP-$complete problem in combinatorial optimization. Now we make a restriction on the graph-searching problem that an edge is closed as soon as it is searched and need not be searched again. The problem arises from epidemic prevention,  the maintenance and repairing of pipeline networks, etc. This restrictive graph-searching problem can be transformed into the elimination cutwidth problem for graphs. In this paper, a polynomial-time algorithm and some fundamental properties of elimination cutwidth $EC(G)$ are mainly presented. Also, the relations with other parameters (such as treewidth and pathwidth) and some special graph results are discussed.