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Last active Nov 6, 2020
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Graceful Tree Conjecture
license: gpl-3.0

The graceful tree conjecture states that every tree has a graceful label. Although proven for some simple cases, the general case remains open.

This program can be used to draw a tree and generate all graceful labels (except for certain trivial variations) for the tree. The visualization also shows the adjacency matrix with respect to the graceful label.

Instructions:

Build tree by clicking on vertices to add leafs to it. Once the desired tree is built, click on 'Generate labels' button to find all graceful labels. The labels shown are unique with respect to adjacency matrix. This reduces many trivial variations in labels, for example if two branches from a vertex are isomorphic, then their labels can be swapped to get another graceful label. Also if (i) is a gaceful labeling, so is (n-i), one of these labelings have 0 and n-1 adjacent. Only labels where 0 and n-1 are adjacent are shown.

<!DOCTYPE html>
<meta charset="utf-8">
<style>
body{
width:960px;
margin:10px auto;
}
circle{
fill:white;
stroke:steelblue;
stroke-width:2px;
}
line{
stroke:grey;
stroke-width:3px;
}
.incRect{
stroke:grey;
shape-rendering:crispEdges;
}
#incMatx text{
text-anchor:middle;
cursor:default;
}
#treesvg g text:hover, #treesvg g circle:hover{
cursor:pointer;
}
#navdiv{
background:#555;
}
#treesvg{
border:1px solid grey;
}
#labelpos{
color:white;
}
#navdiv button, #navdiv textarea{
vertical-align:middle;
}
#g_labels text{
text-anchor:middle;
}
#g_elabels text{
text-anchor:middle;
fill:red;
font-weight:bold;
}
</style>
<body>
<script src="http://d3js.org/d3.v3.min.js"></script>
<script>
function tree(){
var svgW=958, svgH =460, vRad=12, tree={cx:300, cy:30, w:40, h:70};
tree.vis={v:0, l:'?', p:{x:tree.cx, y:tree.cy},c:[]};
tree.size=1;
tree.glabels =[];
tree.incMatx =[];
tree.incX=500, tree.incY=30, tree.incS=20;
tree.getVertices = function(){
var v =[];
function getVertices(t,f){
v.push({v:t.v, l:t.l, p:t.p, f:f});
t.c.forEach(function(d){ return getVertices(d,{v:t.v, p:t.p}); });
}
getVertices(tree.vis,{});
return v.sort(function(a,b){ return a.v - b.v;});
}
tree.getEdges = function(){
var e =[];
function getEdges(_){
_.c.forEach(function(d){ e.push({v1:_.v, l1:_.l, p1:_.p, v2:d.v, l2:d.l, p2:d.p});});
_.c.forEach(getEdges);
}
getEdges(tree.vis);
return e.sort(function(a,b){ return a.v2 - b.v2;});
}
tree.addLeaf = function(_){
function addLeaf(t){
if(t.v==_){ t.c.push({v:tree.size++, l:'?', p:{},c:[]}); return; }
t.c.forEach(addLeaf);
}
addLeaf(tree.vis);
reposition(tree.vis);
if(tree.glabels.length != 0){
tree.glabels =[]
relabel(
{
lbl:d3.range(0, tree.size).map(function(d){ return '?';}),
incMatx:d3.range(0,tree.size-1).map(function(){ return 0;})
});
d3.select("#labelnav").style('visibility','hidden');
}
else tree.incMatx = d3.range(0,tree.size-1).map(function(){ return 0;});
redraw();
}
tree.gracefulLabels = function(){
tree.glabels =[], v = tree.getVertices();
var vlbls =[], elbls=[];
gracefulLbl = function(c){
if(c == tree.size) {
var lbl = {lbl:vlbls.map(function(_){return _;}) };
relabel(lbl);
updateIncMatx();
var incMatx = tree.incMatx.map(function(_){ return _; });
if( (tree.incMatx[0] & 2)>> 1 ==1 && tree.glabels.every(function(d){ return d.incMatx.toString() != incMatx.toString(); })){
lbl.incMatx = incMatx;
tree.glabels.push(lbl);
}
return;
}
d3.range(0, tree.size)
.filter(function(d){ return (vlbls.indexOf(d) ==-1) &&(elbls.indexOf(Math.abs(vlbls[v[c].f.v] - d)) == -1);})
.forEach(function(d){
vlbls[c]=d;
elbls[c]=Math.abs(vlbls[v[c].f.v] - d);
gracefulLbl(c+1);
delete vlbls[c];
delete elbls[c];
});
}
d3.range(0, tree.size).forEach(function(d){ vlbls =[d]; elbls=[]; gracefulLbl(1); });
tree.showLabel(1);
d3.select("#labelpos").text(tree.currLbl+'/'+tree.glabels.length);
d3.select("#labelnav").style('visibility','visible');
}
updateIncMatx = function(){
var n = tree.size-1;
tree.incMatx = d3.range(0,tree.size-1).map(function(){return 0;});
updateIncMatxl = function(t){
t.c.forEach(function(c){
t.l < c.l ? tree.incMatx[t.l]= tree.incMatx[t.l] | (1<<(n-c.l)) : tree.incMatx[c.l]= tree.incMatx[c.l] | (1<<(n-t.l));
updateIncMatxl(c);
});
}
updateIncMatxl(tree.vis);
}
getIncMatxRow = function(i){
return d3.range(0,tree.size-1-i).map(function(d,j){ var n=tree.size-2-i-j; return (tree.incMatx[i] & 1<<n)>>n; });
}
tree.showLabel = function(i){
if(i >tree.glabels.length || i < 1){ alert('invalid label position'); return; }
relabel(tree.glabels[i-1]);
redraw();
tree.currLbl = i;
d3.select("#labelpos").text(tree.currLbl+'/'+tree.glabels.length);
}
relabel = function(lbl){
function relbl(t){ t.l=lbl.lbl[t.v]; t.c.forEach(relbl); }
relbl(tree.vis);
tree.incMatx = lbl.incMatx;
}
redraw = function(){
var edges = d3.select("#g_lines").selectAll('line').data(tree.getEdges());
edges.transition().duration(500)
.attr('x1',function(d){ return d.p1.x;}).attr('y1',function(d){ return d.p1.y;})
.attr('x2',function(d){ return d.p2.x;}).attr('y2',function(d){ return d.p2.y;})
edges.enter().append('line')
.attr('x1',function(d){ return d.p1.x;}).attr('y1',function(d){ return d.p1.y;})
.attr('x2',function(d){ return d.p1.x;}).attr('y2',function(d){ return d.p1.y;})
.transition().duration(500)
.attr('x2',function(d){ return d.p2.x;}).attr('y2',function(d){ return d.p2.y;});
var circles = d3.select("#g_circles").selectAll('circle').data(tree.getVertices());
circles.transition().duration(500).attr('cx',function(d){ return d.p.x;}).attr('cy',function(d){ return d.p.y;});
circles.enter().append('circle').attr('cx',function(d){ return d.f.p.x;}).attr('cy',function(d){ return d.f.p.y;}).attr('r',vRad)
.on('click',function(d){return tree.addLeaf(d.v);})
.transition().duration(500).attr('cx',function(d){ return d.p.x;}).attr('cy',function(d){ return d.p.y;});
var labels = d3.select("#g_labels").selectAll('text').data(tree.getVertices());
labels.text(function(d){return d.l;}).transition().duration(500)
.attr('x',function(d){ return d.p.x;}).attr('y',function(d){ return d.p.y+5;});
labels.enter().append('text').attr('x',function(d){ return d.f.p.x;}).attr('y',function(d){ return d.f.p.y+5;})
.text(function(d){return d.l;}).on('click',function(d){return tree.addLeaf(d.v);})
.transition().duration(500)
.attr('x',function(d){ return d.p.x;}).attr('y',function(d){ return d.p.y+5;});
var elabels = d3.select("#g_elabels").selectAll('text').data(tree.getEdges());
elabels
.attr('x',function(d){ return (d.p1.x+d.p2.x)/2+(d.p1.x < d.p2.x? 8: -8);}).attr('y',function(d){ return (d.p1.y+d.p2.y)/2;})
.text(function(d){return tree.glabels.length==0? '': Math.abs(d.l1 -d.l2);});
elabels.enter().append('text')
.attr('x',function(d){ return (d.p1.x+d.p2.x)/2+(d.p1.x < d.p2.x? 8: -8);}).attr('y',function(d){ return (d.p1.y+d.p2.y)/2;})
.text(function(d){return tree.glabels.length==0? '': Math.abs(d.l1 -d.l2);});
d3.select('#incMatx').selectAll(".incrow").data(tree.incMatx)
.enter().append('g').attr('class','incrow');
d3.select('#incMatx').selectAll(".incrow").selectAll('.incRect')
.data(function(d,i){ return getIncMatxRow(i).map(function(v,j){return {y:i, x:j, f:v};})})
.enter().append('rect').attr('class','incRect');
d3.select('#incMatx').selectAll('.incRect')
.attr('x',function(d,i){ return (d.x+d.y)*tree.incS;}).attr('y',function(d,i){ return d.y*tree.incS;})
.attr('width',function(){ return tree.incS;}).attr('height',function(){ return tree.incS;})
.attr('fill',function(d){ return d.f == 1? 'black':'white'});
d3.select("#incMatx").selectAll('.incrowlabel').data(d3.range(0,tree.size)).enter()
.append('text').attr('class','incrowlabel');
d3.select("#incMatx").selectAll('.incrowlabel').text(function(d){ return d;})
.attr('x',function(d,i){ return (i-0.5)*tree.incS}).attr('y',function(d,i){ return (i+0.8)*tree.incS});
}
getLeafCount = function(_){
if(_.c.length ==0) return 1;
else return _.c.map(getLeafCount).reduce(function(a,b){ return a+b;});
}
reposition = function(v){
var lC = getLeafCount(v), left=v.p.x - tree.w*(lC-1)/2;
v.c.forEach(function(d){
var w =tree.w*getLeafCount(d);
left+=w;
d.p = {x:left-(w+tree.w)/2, y:v.p.y+tree.h};
reposition(d);
});
}
initialize = function(){
d3.select("body").append("div").attr('id','navdiv');
d3.select("#navdiv").append("button").attr('type','button').text('Generate labels')
.on('click',function(d){return tree.gracefulLabels();});
d3.select("#navdiv").append("nav").attr('id','labelnav').style('display','inline-block').style('visibility','hidden');
d3.select("#labelnav").append("button").attr('type','button').text('<').attr('id','prevlabel')
.on('click',function(d){return tree.showLabel(tree.currLbl == 1? tree.glabels.length: tree.currLbl-1);});
d3.select("#labelnav").append("text").text('').attr('id','labelpos');
d3.select("#labelnav").append("button").attr('type','button').text('>').attr('id','nextlabel')
.on('click',function(){return tree.showLabel(tree.currLbl == tree.glabels.length? 1: tree.currLbl+1);});
d3.select("body").append("svg").attr("width", svgW).attr("height", svgH).attr('id','treesvg');
d3.select("#treesvg").append('g').attr('id','g_lines').selectAll('line').data(tree.getEdges()).enter().append('line')
.attr('x1',function(d){ return d.p1.x;}).attr('y1',function(d){ return d.p1.y;})
.attr('x2',function(d){ return d.p2.x;}).attr('y2',function(d){ return d.p2.y;});
d3.select("#treesvg").append('g').attr('id','g_circles').selectAll('circle').data(tree.getVertices()).enter()
.append('circle').attr('cx',function(d){ return d.p.x;}).attr('cy',function(d){ return d.p.y;}).attr('r',vRad)
.on('click',function(d){return tree.addLeaf(d.v);});
d3.select("#treesvg").append('g').attr('id','g_labels').selectAll('text').data(tree.getVertices()).enter().append('text')
.attr('x',function(d){ return d.p.x;}).attr('y',function(d){ return d.p.y+5;}).text(function(d){return d.l;})
.on('click',function(d){return tree.addLeaf(d.v);});
d3.select("#treesvg").append('g').attr('id','g_elabels').selectAll('text').data(tree.getEdges()).enter().append('text')
.attr('x',function(d){ return (d.p1.x+d.p2.x)/2+(d.p1.x < d.p2.x? 8: -8);}).attr('y',function(d){ return (d.p1.y+d.p2.y)/2;})
.text(function(d){return tree.glabels.length==0? '': Math.abs(d.l1 -d.l2);});
d3.select("body").select("svg").append('g').attr('transform',function(){ return 'translate('+tree.incX+','+tree.incY+')';})
.attr('id','incMatx').selectAll('.incrow')
.data(tree.incMatx.map(function(d,i){ return {i:i, r:d};})).enter().append('g').attr('class','incrow');
d3.select("#incMatx").selectAll('.incrowlabel').data(d3.range(0,tree.size)).enter()
.append('text').attr('class','incrowlabel').text(function(d){ return d;})
.attr('x',function(d,i){ return (i-0.5)*tree.incS}).attr('y',function(d,i){ return (i+.8)*tree.incS});
tree.addLeaf(0);
tree.addLeaf(0);
}
initialize();
return tree;
}
var tree= tree();
</script>
@macliems

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@macliems macliems commented Dec 16, 2014

Wonderful! I am trying to understand your code and want to accompish two things: 1) set the nodes further apart from each other (for somewhat longer labels) and 2) give the user the possibility to add a leaf with a custom label. I don't need the conjecture... I would be very grateful for some directions how to do this!

@jenia

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@jenia jenia commented Sep 24, 2016

What is the license for this program?

@Alvynskio

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@Alvynskio Alvynskio commented Mar 8, 2018

Thank you for this! I am in an Undergraduate Research Group at my university also working on the Graceful Tree Conjecture and I am also trying to understand your code. I'm sure I just need a little more time with it. Thanks again.

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