Edge Elimination and Weighted Graph Classes

Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs

“…In both cases (a, b, x, y) is an induced P 4 in G + x y (see Fig. 3(iii)) and we find a and b in time O (1). Similar to the first case, ay is the only non-edge of this path that is degree-lower than x y and the insertion of ay turns the P 4 into a C 4 in any G + x y + F. Now assume that i = 1 and let a ∈ I 1 and b ∈ U 1 .…”

“…To compute theses partitions we only have to remove a constant number of elements from their corresponding lists and put them in lists that are predecessors or successors of the old lists. Therefore, these operations can be done in O (1).…”

“…It follows from the structure of the chain graphs, as given above, that partition P i is a chain partition of graph G + x y if Condition i holds. Furthermore, all partitions can be constructed in time O (1).…”

“…Beisegel et al [1] showed that for any threshold graph (chain graph) and a special partition (E 1 , . .…”

“…While for the studied graph classes the modification of sets of vertices is a straightforward extension of the previous results, the modification of sets of edges needs a different approach which leads to a running time of O(min{k log k, n + k}), where k is the size of the modified edge set. This approach uses special edge elimination schemes introduced by Beisegel et al [1].…”