Extended Path Partition Conjecture for Semicomplete and Acyclic Compositions

11/18/2021
by   Jiangdong Ai, et al.
0

Let D be a digraph and let λ(D) denote the number of vertices in a longest path of D. For a pair of vertex-disjoint induced subdigraphs A and B of D, we say that (A,B) is a partition of D if V(A)∪ V(B)=V(D). The Path Partition Conjecture (PPC) states that for every digraph, D, and every integer q with 1≤ q≤λ(D)-1, there exists a partition (A,B) of D such that λ(A)≤ q and λ(B)≤λ(D)-q. Let T be a digraph with vertex set {u_1,…, u_t} and for every i∈ [t], let H_i be a digraph with vertex set {u_i,j_i j_i∈ [n_i]}. The composition Q=T[H_1,… , H_t] of T and H_1,…, H_t is a digraph with vertex set {u_i,j_i i∈ [t], j_i∈ [n_i]} and arc set A(Q)=∪^t_i=1A(H_i)∪{u_i,j_iu_p,q_p u_iu_p∈ A(T), j_i∈ [n_i], q_p∈ [n_p]}. We say that Q is acyclic (semicomplete, respectively) if T is acyclic (semicomplete, respectively). In this paper, we introduce a conjecture stronger than PPC using a property first studied by Bang-Jensen, Nielsen and Yeo (2006) and show that the stronger conjecture holds for wide families of acyclic and semicomplete compositions.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset