Cliques of complementary graph

Question

"If S is a maximal independent set in some graph, it is a maximal clique or maximal complete subgraph in the complementary graph." from Wikipedia

Does this mean, given a graph G, if it is bipartite, then it's complement G' will have 2 maximal cliques? With, Union of vertices in these two cliques = set of all vertices in G?

example -
image

Here, Gc is bipartite, we can color it with 2 colors, and the independent groups are - {a, b, c} and {d, e}.

And so graph G (it's complement) have 2 maximal cliques {a, b, c} and {d, e}, whose union is all vertices.
So, is it true, for every graph??

Answer 1

Yes, if the vertex set of a graph $G$ can be partitioned into two independent sets, then the vertex set of the complete graph $G^c$ can be partitioned into two cliques (and conversely). More generally, for any subset $W \subseteq V(G)$, $W$ is an independent set of $G$ iff $W$ is a clique of $G^c$. And $W$ is a maximal independent set of $G$ iff $W$ is a maximal clique of $G^c$.

Answer 2

Suppose that $G$ is bipartite, with vertex set $V=V_0\cup V_1$, where $V_0$ and $V_1$ are the two parts. Then $V_0$ and $V_1$ are independent sets, so they will be cliques in the complement of $G$. However, they may not be maximal independent sets in $G$, so they may not be maximal cliques in the complement of $G$. Let $W_0$ be the set of vertices in $V_0$ that are adjacent to some vertex of $V_1$, and let $W_1$ be the set of vertices in $V_1$ that are adjacent to some vertex of $V_0$. Let $I=V\setminus(W_0\cup W_1)$; $I$ contains the isolated vertices of $G$. Then $W_0\cup I$ and $W_1\cup I$ are maximal independent sets in $G$ and maximal cliques in the complement of $G$.