Decomposition of complete graphs into paths and stars

Multi Party Computation Motivated by the Birthday Problem

“…There are polynomial time algorithms that creates cover by trees, K 1,k or P 4 with overlap 2 [2]. In [17] it is proved that optimal covering is polynomial with S k and P k graphs. Covering a graph with complete bipartite subgraphs, but not with a fixed size is discussed in [12].…”

“…Theorem 1 (Theorem B. from [17]). Let p and q nonnegative integers, let n and k be positive integers such that n ≥ 4k and k(p+q) = n 2 , and let one of the following conditions hold: (1) k is even and p ≥ k 2 , (2) k is odd and p ≥ k. Then there exists a decomposition of K n into p copies of P k+1 and q copies of S k+1 .…”

Decomposition of the complete bipartite multigraph into cycles and stars

“…Abueida, Clark, and Leach [1] and Abueida and Hampson [6] considered the existence of decompositions of K n − F for the graph-pairs of order 4 and 5, respectively, where F is a Hamiltonian cycle, a 1-factor, or a graph that is 1-regular except for one vertex of degree 0. Furthermore, Shyu [24] investigated the problem of decomposing K n into paths and stars with k edges, giving a necessary and sufficient condition for k = 3. In [23,25], Shyu considered the existence of a decomposition of K n into paths and cycles with k edges, giving a necessary and sufficient condition for k ∈ {3, 4}.…”

Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars

“…In [23], Priyadharsini and Muthusamy gave necessary and sufficient conditions for the existence of the ( , )-multidecomposition of where , ∈ { , −1 , −1 }. Furthermore, Shyu [24] investigated the problem of decomposing into -paths and -stars, and gave a necessary and sufficient condition for = 3. In [25], Shyu considered the existence of a decomposition of into -paths and -cycles and established a necessary and sufficient condition for = 4.…”