Fuzzy Partial-Order Relations for Intervals and Interval Weighted Graphs

Weighted graphs have been broadly employed in various kinds of applications. Weights associated with edges in a graph are constants mostly in the literature. However, in real world applications, these weights may vary within ranges rather than fixed values. To model such kind of uncertainty or variability, we propose interval-valued weighted graphs in this study. In solving practical graph applications such as finding shortest paths and minimum spanning trees for interval weighted graphs, it is necessary to be able to compare interval valued weights. However, two general intervals can not be ordered reasonably in binary logic. In this paper, we establish fuzzy partial-order relations for intervals. These relations are continuous, except only at a single point in a special case. After studying the properties of the fuzzy partial order relations, we applied the interval partial order to extend well-known shortest path and minimum spanning tree algorithms for interval weighted graphs.