我想在不存在欧拉循环的图中解决中国邮递员问题。所以基本上我正在寻找图中的一条路径，该路径恰好访问每个边一次，并在同一节点处开始和结束。当且仅当每个节点具有相同数量的进入和离开图的边时，图才会有欧拉循环。显然我的图表没有。
我发现欧拉化（制作欧拉图）可以解决我的问题LINK。任何人都可以建议一个脚本来向图中添加重复的边，以便生成的图没有奇数度的顶点（因此具有欧拉电路）？
这是我的例子：
require(igraph)
require(graph)
require(eulerian)
require(GA)
g1 <- graph(c(1,2, 1,3, 2,4, 2,5, 1,5, 3,5, 4,7, 5,7, 5,8, 3,6, 6,8, 6,9, 9,11, 8,11, 8,10, 8,12, 7,10, 10,12, 11,12), directed = FALSE)
mat <- get.adjacency(g1)
mat <- as.matrix(mat)
rownames(mat) <- LETTERS[1:12]
colnames(mat) <- LETTERS[1:12]
g2 <- as(graphAM(adjMat=mat), "graphNEL")
hasEulerianCycle(g2)
有趣的问题。
您在上面的代码中建议的图表可以具有重复项，从而可以创建欧拉循环。我在下面提供的函数尝试添加最少量的重复边，但如果需要，也可以通过添加新链接轻松破坏图形结构。
你可以运行：
eulerian.g1 <- make.eulerian(g1)$graph
检查该函数对您的图表做了什么：
make.eulerian(g1)$info
请记住：
g1图可以构成欧拉循环。想象一下，例如我的函数向后循环图的顶点。也可以看看this关于概率计算的优秀答案：
这是我的完整脚本中的函数，您可以开箱即用：
library(igraph)
# You asked about this graph
g1 <- graph(c(1,2, 1,3, 2,4, 2,5, 1,5, 3,5, 4,7, 5,7, 5,8, 3,6, 6,8, 6,9, 9,11, 8,11, 8,10, 8,12, 7,10, 10,12, 11,12), directed = FALSE)
# Make a CONNECTED random graph with at least n nodes
connected.erdos.renyi.game <- function(n,m){
graph <- erdos.renyi.game(n,m,"gnm",directed=FALSE)
graph <- delete_vertices(graph, (degree(graph) == 0))
}
# This is a random graph
g2 <- connected.erdos.renyi.game(n=12, m=16)
make.eulerian <- function(graph){
# Carl Hierholzer (1873) had explained how eulirian cycles exist for graphs that are
# 1) connected, and 2) contain only vertecies with even degrees. Based on this proof
# the posibility of an eulerian cycle existing in a graph can be tested by testing
# on these two conditions.
#
# This function assumes a connected graph.
# It adds edges to a graph to ensure that all nodes eventuall has an even numbered. It
# tries to maintain the structure of the graph by primarily adding duplicates of already
# existing edges, but can also add "structurally new" edges if the structure of the
# graph does not allow.
# save output
info <- c("broken" = FALSE, "Added" = 0, "Successfull" = TRUE)
# Is a number even
is.even <- function(x){ x %% 2 == 0 }
# Graphs with an even number of verticies with uneven degree will more easily converge
# as eulerian.
# Should we even out the number of unevenly degreed verticies?
search.for.even.neighbor <- !is.even(sum(!is.even(degree(graph))))
# Loop to add edges but never to change nodes that have been set to have even degree
for(i in V(graph)){
set.j <- NULL
#neighbors of i with uneven number of edges are good candidates for new edges
uneven.neighbors <- !is.even(degree(graph, neighbors(graph,i)))
if(!is.even(degree(graph,i))){
# This node needs a new connection. That edge e(i,j) needs an appropriate j:
if(sum(uneven.neighbors) == 0){
# There is no neighbor of i that has uneven degree. We will
# have to break the graph structure and connect nodes that
# were not connected before:
if(sum(!is.even(degree(graph))) > 0){
# Only break the structure if it's absolutely nessecary
# to force the graph into a structure where an euclidian
# cycle exists:
info["Broken"] <- TRUE
# Find candidates for j amongst any unevenly degreed nodes
uneven.candidates <- !is.even(degree(graph, V(graph)))
# Sugest a new edge between i and any node with uneven degree
if(sum(uneven.candidates) != 0){
set.j <- V(graph)[uneven.candidates][[1]]
}else{
# No candidate with uneven degree exists!
# If all edges except the last have even degrees, thith
# function will fail to make the graph eulerian:
info["Successfull"] <- FALSE
}
}
}else{
# A "structurally duplicated" edge may be formed between i one of
# the nodes of uneven degree that is already connected to it.
# Sugest a new edge between i and its first neighbor with uneven degree
set.j <- neighbors(graph, i)[uneven.neighbors][[1]]
}
}else if(search.for.even.neighbor == TRUE & is.null(set.j)){
# This only happens once (probably) in the beginning of the loop of
# treating graphs that have an uneven number of verticies with uneven
# degree. It creates a duplicate between a node and one of its evenly
# degreed neighbors (if possible)
info["Added"] <- info["Added"] + 1
set.j <- neighbors(graph, i)[ !uneven.neighbors ][[1]]
# Never do this again if a j is correctly set
if(!is.null(set.j)){search.for.even.neighbor <- FALSE}
}
# Add that a new edge to alter degrees in the desired direction
# OBS: as.numeric() since set.j might be NULL
if(!is.null(set.j)){
# i may not link to j
if(i != set.j){
graph <- add_edges(graph, edges=c(i, set.j))
info["Added"] <- info["Added"] + 1
}
}
}
# return the graph
(list("graph" = graph, "info" = info))
}
# Look at what we did
eulerian <- make.eulerian(g1)
eulerian$info
g <- eulerian$graph
par(mfrow=c(1,2))
plot(g1)
plot(g)