A graph G is said to be hamiltonian path saturated (HPS for short), if G has no hamiltonian path but any addition of a new edge in G creates a hamiltonian path in G. It is known that an HPS graph of order n has size at most (n-1 2) and, for n ≥ 6, the only HPS graph of order n and size (n-1 2) is Kn-1 ∪ K1. Denote by sat(n, HP) the minimum size of an HPS graph of order n. We prove that sat(n, HP) ≥ ⌊ (3n-1)/2 ⌋ - 2. Using some properties of Isaacs' snarks we give, for every n ≥ 54, an HPS graph Gn of order n and size ⌊ (3n - 1)/2 ⌋. This proves sat(n, HP) ≤ ⌊ (3n - 1)/2 ⌋ for n ≥ 54. We also consider m-path cover saturated graphs and Pm-saturated graphs with small size.

Hamiltonian shortage, path partitions of vertices, and matchings in a graph

Hamiltonian path saturated graphs with small size

Given a graph G = (V,E), HCN(L(G)) is the minimum number of edges to be added to its line graph L(G) to make L(G) Hamiltonian. This problem is known to be NP-hard for general graphs, whereas a O(|V|) algorithm exists when G is a tree. In this paper a linear algorithm for finding HCN(L(G)) when G is a cactus is proposed.

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

Given a graph G=(V,E), the Hamiltonian completion number of G, HCN(G), is the minimum number of edges to be added to G to make it Hamiltonian. This problem is known to be \(\mathcal{NP}\) -hard even when G is a line graph. In this paper, local search algorithms for finding HCN(G) when G is a line graph are proposed. The adopted approach is mainly based on finding a set of edge-disjoint trails that dominates all the edges of the root graph of G. Extensive computational experiments conducted on a wide set of instances allow to point out the behavior of the proposed algorithms with respect to both the quality of the solutions and the computation time.

background

Local search algorithms for finding the Hamiltonian completion number of line graphs

A condition for a graph to contain k-matching

m_Path Cover Saturated Graphs

/pdf/hamiltonian-shortage-path-partitions-of-vertices-and-3ynsshf2c1.pdf

Hamiltonian shortage, path partitions of vertices, and matchings in a graph

Citations

Hamiltonian path saturated graphs with small size

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

Local search algorithms for finding the Hamiltonian completion number of line graphs

A condition for a graph to contain k-matching

m_Path Cover Saturated Graphs

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