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48
Supereulerian graphs: A survey
 J. Graph Theory
, 1992
"... A graph is supereulerian if it has a spanning eulerian subgraph. There is a reduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a grap ..."
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Cited by 30 (4 self)
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A graph is supereulerian if it has a spanning eulerian subgraph. There is a reduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a graph. We outline the research on supereulerian graphs, the reduction method and its applications. 1. Notation We follow the notation of Bondy and Murty [22], with these exceptions: a graph has no loops, but multiple edges are allowed; the trivial graph K1 is regarded as having infinite edgeconnectivity; and the symbol E will normally refer to a subset of the edge set E(G) of a graph G, not to E(G) itself. The graph of order 2 with 2 edges is called a 2cycle and denoted C2. Let H be a subgraph of G. The contraction G/H is the graph obtained from G by contracting all edges of H and deleting any resulting loops. For a graph G, denote O(G) = {odddegree vertices of G}. A graph with O(G) = ∅ is called an even graph. A graph is eulerian if it is connected and even. We call a graph G supereulerian if G has a spanning eulerian subgraph. Regard K1 as supereulerian. Denote SL = {supereulerian graphs}. 1 Let G be a graph. The line graph of G (called an edge graph in [22]) is denoted L(G), it has vertex set E(G), where e, e ′ ∈ E(G) are adjacent vertices in L(G) whenever e and e ′ are adjacent edges in G. Let S be a family of graphs, let G be a graph, and let k ≥ 0 be an integer. If there is a graph G0 ∈ S such that G can be obtained from G0 by removing at most k edges, then G is said to be at most k edges short of being in S. For a graph G, we write F (G) = k if k is the least nonnegative integer such that G is at most k edges short of having 2 edgedisjoint spanning trees. 2.
MATCHING, EULER TOURS AND THE CHINESE POSTMAN
 MATHEMATICAL PROGRAMMING 5 (1973) 88124. NORTHHOLLAND PUBLISHING COMPANY
, 1973
"... The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more general bm ..."
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Cited by 29 (0 self)
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The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more general bmatching blossom algorithm. Algorithms for finding Euler tours and related problems are also discussed.
From decision theory to decision aiding methodology (my very personal version of this history and some related reflections)
, 2003
"... ..."
Faster evolutionary algorithms by superior graph representation
 In First IEEE Symposium on Foundations of Computational Intelligence (FOCI2007
, 2007
"... Abstract — We present a new representation for individuals in problems that have cyclic permutations as solutions. To demonstrate its usefulness rigorously, we construct from it a simple randomized local search and a (1+1) evolutionary algorithm for the Eulerian cycle problem. Both have an expected ..."
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Cited by 13 (4 self)
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Abstract — We present a new representation for individuals in problems that have cyclic permutations as solutions. To demonstrate its usefulness rigorously, we construct from it a simple randomized local search and a (1+1) evolutionary algorithm for the Eulerian cycle problem. Both have an expected runtime of Θ(m 2 log(m)), where m denotes the number of edges of the input graph. This clearly beats previous solutions, which all have an expected optimization time of Θ(m 3) or worse (PPSN ’06, CEC ’04). We are optimistic that our representation allows superior solutions also for other cyclic permutation problems. For NPcomplete ones like the TSP, however, other means than theoretical runtime analyses are necessary. I.
Catching the ‘Network Science’ Bug: Insight and Opportunities for the Operations Researchers
 Operations Research
, 2009
"... Accepted for publication by ..."
Adjacency list matchings — an ideal genotype for cycle covers
 In Genetic and Evolutionary Computation Conference (GECCO2007
, 2007
"... We propose and analyze a novel genotype to represent walk and cycle covers in graphs, namely matchings in the adjacency lists. This representation admits the natural mutation operator of adding a random match and possibly also matching the former partners. To demonstrate the strength of this setup, ..."
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Cited by 8 (3 self)
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We propose and analyze a novel genotype to represent walk and cycle covers in graphs, namely matchings in the adjacency lists. This representation admits the natural mutation operator of adding a random match and possibly also matching the former partners. To demonstrate the strength of this setup, we use it to build a simple (1+1) evolutionary algorithm for the problem of finding an Eulerian cycle in a graph. We analyze several natural variants that stem from different ways to randomly choose the new match. Among other insight, we exhibit a (1+1) evolutionary algorithm that computes an Euler tour in a graph with m edges in expected optimization time Θ(m log m). This significantly improves the previous best evolutionary solution having expected optimization time Θ(m2 log m) in the worstcase, but also compares nicely with the runtime of an optimal classical algorithm which is of order Θ(m). A simple coupon collector argument indicates that our optimization time is asymptotically optimal for any randomized search heuristic.
Introduction to the special section on graph algorithms in computer vision
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... ..."
A short note on the history of graph drawing
 IN GD ’01
, 2002
"... The origins of chart graphics (e.g., bar charts and line charts) are well known [30], with the seminal event being the publication of William Playfair’s (17591823) The Commercial and Political Atlas in London in 1786 [26]. However, the origins of graph drawing are not well known. Although Euler ( ..."
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The origins of chart graphics (e.g., bar charts and line charts) are well known [30], with the seminal event being the publication of William Playfair’s (17591823) The Commercial and Political Atlas in London in 1786 [26]. However, the origins of graph drawing are not well known. Although Euler (17071783) is credited with originating graph theory in 1736 [12,20], graph drawings were in limited use centuries before Euler’s time. Moreover, Euler himself does not appear to have made significant use of graph visualizations. Widespread use of graph drawing did not begin until decades later, when it arose in several distinct contexts. In this short note we present a selection of very early graph drawings; note the apparent absence of graph visualization in Euler’s work; and identify some early innovators of modern graph drawing.