Task 1 Consider a partial ordering ≺ of a finite set X, and let G be the corresponding comparability graph. • What do the independent sets, the cliques, the proper colorings of G, and the proper colorings of the complement of G correspond to in the partial ordering ≺? • Using the weak perfect graph theorem and the fact that comparability graphs are perfect, prove Dilworth theorem: The maximum size of an antichain in ≺ is equal to the minimum number of chains in ≺ whose union is X. Task 2 Let π be a permutation of f1; : : : ; ng and let Gπ be the graph with the vertex set f1; : : : ; ng where for 1 ≤ i < j ≤ n, ij 2 E(Gπ) iff π(i) > π(j). Any graph isomorphic to a graph obtained in this way is called a permutation graph. • Prove that every permutation graph is also the comparability graph of some partial ordering. • Prove that the complement of every permutation graph is also a permu- tation graph. • Show that G is a permutation graph if and only if there exist two linear orderings ≺1 and ≺2 of V (G) such that for every u; v 2 V (G) satisfying u ≺1 v, we have uv 2 E(G) if and only if u ≺2 v. I.e., two vertices are adjacent iff they compare the same in ≺1 and in ≺2. • Suppose that G is the comparability graph of a partial ordering ≺a and G is the comparability graph of a partial ordering ≺b. Show that the relation ≺ on V (G) defined so that u ≺ v iff u ≺a v or u ≺b v is a linear ordering of V (G). 0 • The reverse of a partial ordering ≺b is the partial ordering ≺b such that 0 u ≺b v iff v ≺b u. Note that the comparability graphs of a partial ordering and its reverse are the same, and thus the previous point gives two linear orderings of V (G). • Show that a graph G is a permutation graph if and only if both G and G are comparability graphs. 1 Task 3 The join of two graphs G1 and G2 is obtained from their disjoint union by adding all edges with one end in V (G1) and the other end in V (G2). A cograph is any graph obtained from single-vertex graphs by any sequence of disjoint unions and joins. • Show that if G1 and G2 are perfect, then both their join and their disjoint union are perfect. Conclude that cographs are perfect. • Show that a cograph does not contain P4 (the path with four vertices) as an induced subgraph. • Let v be a vertex of a graph G and suppose that G − v is not connected. Show that one of the following holds: G is not connected, or v is adjacent to all other vertices of G, or G contains P4 as an induced subgraph. • Using the previous point, show by induction on the number of vertices that if G is a graph with at least two vertices not containing P4 as an induced subgraph, then either G or the complement of G is disconnected. • Show that if G does not contain P4 as an induced subgraph, then G is a cograph. Task 4 A graph G is distance-hereditary if for any connected induced subgraph H of G, the distance between any two vertices in H is the same as their distance in G. • Show that G is distance-hereditary if and only if for every u; v 2 E(G), all induced paths in G between u and v have the same length (equal to the distance between u and v in G). • Use the strong perfect graph theorem to show that distance-hereditary graphs are perfect. 2.