こんにちは。
gonum/graph を利用し、グラフの連結成分（＝サブグラフ）内のエッジを列挙してみました。グラフは undirected multigraph で、エッジは複数の lines を持ちます。

DepthFirst.EdgeFilter() へ edgeFunc() を渡し、サブグラフ単位で全ての lineID を列挙させています。

$ go run walkFrom.go
input undirected lines: [[0 1] [1 2] [2 3] [3 0] [0 2] [1 3] [1 0] [4 4] [4 4]]
total 9 lines

lineID   ! edge
----------------
2        0 [2 3]
1        1 [1 2]
5        2 [1 3]
0        3 [0 1]
6        3 [0 1]
3        4 [0 3]
4        5 [0 2]
----------------
8        6 [4 4]
7        6 [4 4]
----------------

walkFrom.go

package main

import (
	"fmt"
	"gonum.org/v1/gonum/graph"
	"gonum.org/v1/gonum/graph/multi"
	"gonum.org/v1/gonum/graph/traverse"
)

var iEdge int

func printLine(lID, mID, nID int64, iEdge int) {
	fmt.Printf("%d        %d [%d %d]\n", lID, iEdge, mID, nID)
}

func printHeader() {
	iEdge = 0
	fmt.Printf("\nlineID   ! edge\n")
	printSeparator()
}

func printSeparator() {
	fmt.Print("----------------\n")
}

func printLines(e graph.Edge) {
	mID, nID := e.From().ID(), e.To().ID()
	if mID > nID {mID, nID = nID, mID}
	l := e.(graph.Lines)
	for l.Next() {
		printLine(l.Line().ID(), mID, nID, iEdge)
	}
	iEdge++
}

func makeWalker(edgeFunc func (graph.Edge)) (walker traverse.DepthFirst) {
	walker.EdgeFilter = func(e graph.Edge) bool {
		if e.From().ID() <= e.To().ID() {
			edgeFunc(e)
		}
		return true
	}
	return walker
}

func walk(g *multi.UndirectedGraph, walker *traverse.DepthFirst, from graph.Node) {
	if !walker.Visited(from) && g.From(from.ID()).Next() {  // neither visited nor isolated node
		walker.Walk(g, from, nil)
		printSeparator()
	}
}

func walkAll(g *multi.UndirectedGraph) {
	printHeader()
	nod := g.Nodes()
	w := makeWalker(printLines)
	for nod.Next() {
		walk(g, &w, nod.Node())
	}
}

func makeGraph() *multi.UndirectedGraph {
	IDMAX := 9
	inputLines := [][]int64{{0, 1}, {1, 2}, {2, 3}, {3, 0}, {0, 2}, {1, 3},  {1, 0}, {4, 4}, {4, 4}}

	g := multi.NewUndirectedGraph()
	for i := 0; i <= IDMAX; i++ {
		g.AddNode(multi.Node(int64(i)))
	}
	fmt.Print("input undirected lines: [")
	for i, e := range inputLines {
		g.SetLine(g.NewLine(g.Node(e[0]), g.Node(e[1])))
	 	fmt.Print(e)
	 	if i < len(inputLines)-1 {fmt.Print(" ")}
	}
	fmt.Printf("]\ntotal %d lines\n", len(inputLines))
	return g
}

func main() {
	g:= makeGraph()
	walkAll(g)
}