On the Burkard–Hammer condition for hamiltonian split graphs

2018

In this paper we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of K1,4-free split graphs in the context of Hamiltonian path problem, and as a consequence, we obtain a polynomial-time algorithm to the Hamiltonian path problem in K1,4-free split graphs. We close this paper with the hardness result: we show that, unless P=NP, Hamiltonian path problem is NP-complete in K1,5-free split graphs by reducing from Hamiltonian cycle problem in K1,5-free split graphs. Thus this paper establishes a “thin complexity line” separating NP-complete instances and polynomial-time solvable instances.

Cornell University - arXiv, 2017

In this paper we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of K1,4-free split graphs in the context of Hamiltonian path problem, and as a consequence, we obtain a polynomial-time algorithm to the Hamiltonian path problem in K1,4-free split graphs. We close this paper with the hardness result: we show that, unless P=NP, Hamiltonian path problem is NP-complete in K1,5-free split graphs by reducing from Hamiltonian cycle problem in K1,5-free split graphs. Thus this paper establishes a "thin complexity line" separating NP-complete instances and polynomial-time solvable instances. Lemma 3 (Chvatal[21]). Let G be a connected graph. If G has a Hamiltonian path, then for every S ⊂ V (G), c(G − S) ≤ |S| + 1. Lemma 4 ([19]). For a connected split graph G with ∆ I = 3, let v ∈ K, d I (v) = 3, and U = N I (v). If G is K 1,4-free, then N (U) = K. Corollary 1 (of Lemma 4). Let G be a connected K 1,4-free split graph with v ∈ K, d I (v) = 3. For every vertex w ∈ K \ {v}, N I (v) ∩ N I (w) = ∅. 2.1 Results on K 1,3-free split graphs Theorem 2. Let G be a connected K 1,3-free split graph. G contains a Hamiltonian path if and only if G has at most 2 vertices u, v ∈ I such that d(u) = 1, and d(v) = 1. Proof. If there exists at least three vertices {u, v, w} ⊆ I such that d(u) = d(v) = d(w) = 1, then clearly G has no Hamiltonian path. For the sufficiency, we see the following cases. Case 1: For every u ∈ I, if d(u) ≥ 2, then G is 2-connected, and by Lemma 2, G has a Hamiltonian cycle. Thus G has a Hamiltonian path. Case 2: If there exists only one vertex u ∈ I with d(u) = 1, then observe that G − u is 2-connected. By Lemma 2, there is a Hamiltonian cycle in G − u, which can be easily extended to a Hamiltonian path in G. Case 3: There exists two vertices u, v ∈ I with d(u) = d(v) = 1. If I = {u, v}, it is easy to see that there is a (u, v)-Hamiltonian path in G. If I = {u, v, w}, then we claim that N (w) ∩ N (u) = ∅ and N (w) ∩ N (v) = ∅. Suppose N (w) ∩ N (u) = ∅, then let N I (u ′) = {u, w}, u ′ ∈ K. Clearly, all the vertices x ∈ K \ {u ′ } are adjacent to w, otherwise {u ′ , u, w, x} induces a K 1,3. It follows that K ∪ {w} is a clique of larger size, contradicting the maximality of K. Similar arguments hold with respect to the vertex v, and hence N (w) ∩ N (v) = ∅. Thus we conclude that ∆ I = 1. From Lemma 1, if |I| > 3, since G is connected, ∆ I = 1. Now we produce a Hamiltonian path in G with |I| ≥ 3 as follows. Let I = {u, v, w 1 ,. .. , w k }, k ≥ 1 such that for all w i , 1 ≤ i ≤ k, d(w i) > 1. Let x i , y i , 1 ≤ i ≤ k be any two elements in N (w i). Since ∆ I = 1, note that for all s, t ∈ I, N (s) ∩ N (t) = ∅. Let P

International Journal of Foundations of Computer Science, 2021

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

European Journal of Combinatorics, 2014

The class of split graphs consists of those graphs G, for which there exist partitions (V 1 , V 2) of their vertex sets V (G) such that G[V 1 ] is an edgeless graph and G[V 2 ] is a complete graph. The classes of edgeless and complete graphs are members of a family L * ≤ , which consists of all graph classes that are induced hereditary and closed under substitution. Graph classes considered in this paper are alike to split graphs. Namely a graph class P is an object of our interest if there exist two graph classes P 1 , P 2 ∈ L * ≤ (not necessarily different) such that for each G ∈ P we can find a partition (V 1 , V 2) of V (G) satisfying G[V 1 ] ∈ P 1 and G[V 2 ] ∈ P 2. For each such class P we characterize all forbidden graphs that are strongly decomposable. The finiteness of families of forbidden graphs for P is analyzed giving a result characterizing classes with a finite number of forbidden graphs. Our investigation confirms, in the class L * ≤ , Zverovich's conjecture describing all induced hereditary graph classes defined by generalized vertex 2-partitions that have finite families of forbidden graphs.

2014

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many minimal chordal obstructions.) We provide (close) upper and lower bounds on the maximum size of a minimal split obstruction.

Discrete Mathematics, 1996

Related to Chvfital's famous conjecture stating that every 2-tough graph is hamiltonian, we study the relation of toughness and hamiltonieity on special classes of graphs. First, we consider properties of graph classes related to hamiltonicity, traceability and toughness concepts and display some algorithmic consequences. Furthermore, we present a polynomial time algorithm deciding whether the toughness of a given split graph is less than one and show that deciding whether the toughness of a bipartite graph is exactly one is coNP-complete. We show that every 3-tough split graph is hamiltonian and that there is a sequence of nonhamiltonian split graphs with toughness converging to 3.

2002

Publikacije Elektrotehni?kog fakulteta - serija: matematika, 2002

We give characterizations of integral graphs in the family of complete split graphs and a few related families of graphs.

Discussiones Mathematicae Graph Theory, 2007

For a graph G of order n we consider the unique partition of its vertex set V (G) = A ∪ B with A = {v ∈ V (G) : d(v) ≥ n/2} and B = {v ∈ V (G) : d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.

Discrete Mathematics, 1993

Shader, B.L., On biclique partitions of the complete graph, Discrete Mathematics 117 (1993) 197-213.