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47
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 31 (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.
From decision theory to decision aiding methodology (my very personal version of this history and some related reflections)
, 2003
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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
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Approximation Schemes for Maximum Cardinality Matching
, 1995
"... Let G = (V; E) be an undirected graph. Given an odd number k = 2l + 1, a matching M is said to be koptimal if it does not admit an augmenting path of length less than or equal to k. We prove jM j jM 3 j(l+1)=(l+2), where M 3 is a maximum cardinality matching. If M is not already (k + 2)optima ..."
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Cited by 4 (0 self)
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Let G = (V; E) be an undirected graph. Given an odd number k = 2l + 1, a matching M is said to be koptimal if it does not admit an augmenting path of length less than or equal to k. We prove jM j jM 3 j(l+1)=(l+2), where M 3 is a maximum cardinality matching. If M is not already (k + 2)optimal, using M , in O(jEj) time we compute a (k + 2)optimal matching. We show that starting with any matching, repeated koptimizations result in an optimal matching in O( p jM 3 jjEj) time. This approximation scheme extends to the minimum weight perfect matching, and the minimum weight uncapacitated perfect 2matching problems over complete graphs, and complete bipartite graphs with edge weights of one and two. In particular we obtain a fast approximation algorithm for the traveling salesman problem over complete graphs with edge weights of one and two.
Effort Flow Analysis: A Methodology for Directed Product Evolution Using Rigid Body and Compliant Mechanisms
, 2002
"... This dissertation presents a systematic design methodology for directed product
evolution that uses both rigid body and compliant mechanisms to facilitate
component combination in the domain of mechanical products. The methodology,
known as effort flow analysis, is based on fundamental tenets from ..."
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Cited by 3 (0 self)
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This dissertation presents a systematic design methodology for directed product
evolution that uses both rigid body and compliant mechanisms to facilitate
component combination in the domain of mechanical products. The methodology,
known as effort flow analysis, is based on fundamental tenets from the Theory of
Mechanics, and Graph Theory. Effort flow analysis uses a semantic network known
as an effort flow diagram to model a product as a connected set of nodes and links.
The nodes represent the components of the product and the links represent the
interfaces between the components. The effort flow diagram is a quasistatic model
of the flow of effort (force or torque in the mechanical domain) as it transits the body
from input to output. In order to capture the effect of the relative motion that occurs
at the interfaces, a basis set for relative motion is developed for effort flow analysis.
The basis consists of 4 possible link type cases, (1) No relative motion at the
interface or away from the interface, (2) Component relative motion that occurs away
from the interface, (3) general Relative motion between components, and (4)
Interface relative motion that occurs only at the interface. These are known as N
Links, CLinks, RLinks, and ILinks respectively. Rigid body combinations are
sought for components connected by NLinks and compliant mechanism
combinations are sought for components connected by the other link types.
Component combination opportunities are sought based on the connection structure
of the effort flow diagram. Guidance for component combination is captured in a set
of 29 productevolution guidelines that are derived from the results of an empirical
study. In addition, an example solution is cataloged for each of the guidelines to
foster designbyanalogy efforts by the designer. Finally, the work presents a novel
design and prototype for a compliant onepiece umbrella frame, which serves as a
proof of concept for the methodology. Possible future pursuits are presented in the
areas of knowledge capture using guidelines and solution examples. In addition, the
possibility of a functiontocomponent matrix for automated concept generation is
considered.