Bandwidth is a fundamental concept in graph theory which has some surprising applications to a class of AI search problems. Graph bandwidth provides a link between the syntactic structure of a constraint satisfaction problem (CSP) and the complexity of the underlying search task. Bandwidth can be used to define a new class of easy CSP’s, namely those that have limited constraint graph bandwidth. These CSP’s can be solved in polynomial time, essentially by divide and conquer. This in turn suggests that bandwidth provides a mathematical measure of the decomposability of a search problem. In addition, bandwidth supplies a measure for comparing different search orderings for a given CSP. Statistical analysis suggests that backtracking with small bandwidth orderings leads to a more efficient search than that obtained under orderings with larger bandwidths. Small bandwidth orderings also limit the pruning that can be done by intelligent backtracking. If small bandwidth orderings are indeed advantageous, then a large number of heuristics developed in numerical analysis to find such orderings may find applicability to solving constraint satisfaction problems.