1

I'm confused in graph cliques whether an edge could be considered as a clique or not as both are adjacent to each other like in attached photo, there is no clique of 3 or more edges.

Example

enter image description here

I considered all the edges as clique as there is no clique with 3 or more vertices.

James Z
  • 12,209
  • 10
  • 24
  • 44
Waleed Ashraf
  • 13
  • 3

1 Answers1

1

Yes, cliques can contain only one vertex. All nodes are by themselves a clique (a 1-clique).

And can contain pairs of nodes connected by an edge. So all edges also describe a clique (clique made of the 2 vertices connected by this edge; so you could, by language extension, say that the edge is a clique. The pair of node surely is). All edges are a 2-clique.

Sure, cliques become interesting only from 3-clique. 1-clique and 2-clique are boring, and just made of single nodes or single edges. But they are cliques anyway.

chrslg
  • 9,023
  • 5
  • 17
  • 31
  • If there is no 3-clique in a graph than all the edges will considered as maximal and maximum clique? – Waleed Ashraf Dec 18 '22 at 13:49
  • @WaleedAshraf To see if you understood, can you list all the cliques, maximal cliques and maximum cliques of this graph? https://i.imgur.com/ygS64PF.png – Stef Dec 18 '22 at 13:51
  • @WaleedAshraf Although note that we typically don't bother listing non-maximal cliques, because every induced subgraph of a clique is always obviously a clique. For instance, if in some graph, abcd is a 4-clique, then obviously abc, bcd, acd, abd are all 3-cliques, ab, ac, ad, bc, bd, cd are all 2-cliques, and a, b, c, d are all 1-clique. So we'll summarise all this by just saying "abcd is a 4-clique" and not bother listing all those subcliques. – Stef Dec 18 '22 at 13:54
  • @WaleedAshraf. Yes, I think so. The paradox. is, I teach this, but nevertheless I am never 100% sure when question in reality just come to definition (it is not a matter of computation of any sort. Just of definition of what is a clique, a maximum clique, etc. A convention). And these are matters in which definition change sometimes with language. But, if the definition is the one I am used to, and if I translate correctly "maximal and maximum" (we don's distinguish in my language. We say "maximal" and "graph maximal"), then, yes, no edges in your example could be extented to 3-clique. – chrslg Dec 18 '22 at 13:56
  • So they are all maximal clique. And since the biggest clique has 2 vertices, then all 2-cliques, that is all edges, are also maximum cliques. – chrslg Dec 18 '22 at 13:58
  • In other words, if I were you, my answer would have been the same for all questions. And, again, I teach this (but I am not very good at it. Sometimes, schools have to find a volunteer to teach some topic nobody is specialist of), but yet, I am never 100% comfortable for what are in reality definition problem (I ceased to ask such definition question to my student a while ago. When I started teaching this, all my exams were looking like this one. Now I try to avoid that, because all question are just definition trap more than understanding evaluation) – chrslg Dec 18 '22 at 14:00
  • Ok now i'm 95% clear about my confussion and i think that is enough. I'm really thankfull to both you for replying me. – Waleed Ashraf Dec 18 '22 at 14:34