Dual Eulerian Properties of Plane Multigraphs

The Recognition of Double Euler Trails in Series-Parallel Networks

“…and an exponential algorithm for finding the DCT N . Several other papers have also explored the use of graph models to find solutions for w x layout 4,10,11,12 . However,using In this paper we study the properties of DET networks and give a linear time algorithm for recognizing DET networks.…”

Finding double Euler trails of planar graphs in linear time [CMOS VLSI circuit design]

“…Whether the DET-V problem can be solved in polynomial time is posed as an open question in [1]. Here, we answer this question in the affirmative, by presenting a linear-time algorithm for the DET-V problem.…”

“…For general TTGs, the DTC-F problem is NP-hard [12,13], which implies that the DTC-V problem for TTGs is also NP-hard. A lineartime algorithm for the DET-F problem was given in [1].…”

“…Lemma 3 (see [1]) Let e = fu; vg be an edge in G and e = fF 1 ; F 2 g its dual edge. Let Q be a maximal double trail in G starting with e. If the direction of e in Q and the direction of e in the dual trail Q are given, then Q is uniquely determined and can be traced out in OjQj time.…”

“…In the dual graph G = V ; E of G, there are two dual vertices s and t corresponding to the outer face of G. The dual edges incident to s (resp., t ) in G correspond to the edges on the left (resp., right) side of G. G is a TTG with source s and sink t [10]. 1 Previously called planar st-graphs.…”