Knowledge

Bound graph

Source 📝

106:, a family of cliques that cover all edges, with the additional property that each clique includes a vertex that does not belong to any other clique in the family. For the bound graph of a given partial order, each clique can be taken to be the subset of elements less than or equal to some given element. A graph that is covered by cliques in this way is the bound graph of a partial order on its vertices, obtained by ordering the unique vertices in each clique as a chain, above all other vertices in that clique. 33: 215: 118: 220: 40: 25: 64: 117:
comprise exactly the same class—any lower bound for ≤ is easily seen to be an upper bound for the
103: 192: 161:
Bergstrand, D.J.; Jones, K.F. (1988). "On upper bound graphs of partially ordered sets".
177: 209: 17: 146:
Lundgren, J.R.; Maybee, J.S. (1983). "A characterization of upper bound graphs".
131:
McMorris, F.R.; Zaslavsky, T. (1982). "Bound graphs of a partially ordered set".
29: 197: 133:
Journal of Combinatorics, Information & System Sciences
39:
is a bound graph if there exists a partial order ≤ on the
102:
The bound graphs are exactly the graphs that have a
8: 109:Bound graphs are sometimes referred to as 24:expresses which pairs of elements of some 196: 47:with the property that for any vertices 7: 185:Electronic Journal of Combinatorics 14: 113:, but the analogously defined 1: 237: 176:Tanenbaum, P.J. (2000). 178:"Bound graph polysemy" 163:Congressus Numerantium 148:Congressus Numerantium 79:and there is a vertex 26:partially ordered set 32:. Rigorously, any 115:lower bound graphs 111:upper bound graphs 121:partial order ≥. 104:clique edge cover 228: 202: 200: 182: 170: 155: 140: 236: 235: 231: 230: 229: 227: 226: 225: 206: 205: 180: 175: 160: 145: 130: 127: 71:if and only if 12: 11: 5: 234: 232: 224: 223: 218: 216:Graph families 208: 207: 204: 203: 172: 171: 157: 156: 142: 141: 126: 123: 13: 10: 9: 6: 4: 3: 2: 233: 222: 219: 217: 214: 213: 211: 199: 198:10.37236/1521 194: 190: 186: 179: 174: 173: 168: 164: 159: 158: 153: 149: 144: 143: 138: 134: 129: 128: 124: 122: 120: 116: 112: 107: 105: 100: 98: 94: 90: 86: 82: 78: 74: 70: 66: 62: 58: 54: 50: 46: 42: 38: 35: 31: 27: 23: 19: 221:Order theory 188: 184: 166: 162: 151: 147: 136: 132: 114: 110: 108: 101: 96: 92: 88: 84: 80: 76: 72: 68: 60: 56: 52: 48: 44: 36: 21: 18:graph theory 15: 30:upper bound 22:bound graph 210:Categories 169:: 185–193. 154:: 189–193. 139:: 134–138. 125:References 83:such that 191:: #R43. 41:vertices 28:have an 63:is an 181:(PDF) 34:graph 119:dual 91:and 65:edge 51:and 20:, a 193:doi 67:of 55:of 43:of 16:In 212:: 187:. 183:. 167:66 165:. 152:40 150:. 135:. 99:. 95:≤ 87:≤ 75:≠ 61:uv 59:, 201:. 195:: 189:7 137:7 97:w 93:v 89:w 85:u 81:w 77:v 73:u 69:G 57:G 53:v 49:u 45:G 37:G

Index

graph theory
partially ordered set
upper bound
graph
vertices
edge
clique edge cover
dual
"Bound graph polysemy"
doi
10.37236/1521
Categories
Graph families
Order theory

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.