The spanning subgraphs of eulerian graphs

Abstract

It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K ( 2 m + 1,2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to such graphs in order to get eulerian graphs.

Efficient Algorithms for Eulerian Extension

“…In fact, the subsequent Proposition 4 implies that unweighted RP is solvable in polynomial time if the input digraph is complete. More precisely, we can solve MEE for directed inputs by a straightforward greedy strategy much like the algorithm known for undirected multigraphs [3].…”

“…Proposition 4 (See [3]). Multigraph Eulerian Extension on directed and undirected multigraphs can be solved in O(n + m) time.…”

“…Lesniak and Oellermann [13] presented an overview of undirected Eulerian graphs. The unweighted and undirected extension problems for graphs and multigraphs were already discussed by Boesch et al [3], who developed a linear time algorithm for the multigraph case and a matching based algorithm for the graph case. Recently, Höhn et al [11] initiated a study of Eulerian extension problems applied to sequencing problems.…”

A New View on Rural Postman Based on Eulerian Extension and Matching

“…However, c influences the degree of the polynomial in the running time of Frederickson's algorithm. To date, it is open whether this is unavoidable 4 or whether RP can be solved in f (c) · n O(1) time for some function f . In other words, it remains open whether RP (and EE) is fixed-parameter tractable with respect to the parameter c [10].…”

“…EE has been shown to be polynomial-time solvable in some special cases [4,10,22,24]. Dorn et al [10] also proved that EE is fixed-parameter tractable with respect to the parameter "number of arcs in the sought Eulerian extension".…”