A closer look at Hamiltonicity and domination through the lens of diameter and convexity

A bipartite graph G(X, Y) is called a star-convex bipartite graph with convexity on X if there is an associated star T(X, F), such that for each vertex in Y, its neighborhood in X induces a subtree in T. A graph G is said to be a split graph if G can be partitioned into a clique (K) and an independent set (I). The objective of this study is twofold: (i) to strengthen the complexity results presented in Chen et al. (J Comb Optim 32(1):95–110, 2016) for the Hamiltonian cycle (HCYCLE), the Hamiltonian path (HPATH), and the Domination (DS) problems on star-convex bipartite graphs (ii) to reinforce the results of Müller (Discret Math 156(1–3):291–298, 1996) for HCYCLE, and HPATH on split graphs by introducing a convex ordering on one of the partitions (K or I). As part of our fine-grained analysis study with the diameter being the parameter, we first show that the diameter of star-convex bipartite graphs is at most six. Next, we observe that the reduction instances of Chen et al. (J Comb Optim 32(1):95–110, 2016) are star-convex bipartite graphs with at most diameter 4, and hence HCYCLE and HPATH are NP-complete on star-convex bipartite graphs with at most diameter 4. We strengthen this result and establish the following results on star-convex bipartite graphs: (i) HCYCLE is NP-complete for diameter 3, and polynomial-time solvable for diameters 2, 5, and 6 (a transformation in complexity: P to NPC to P) (ii) HPATH is polynomial-time solvable for diameter 2, and NP-Complete, otherwise (a dichotomy). Further, with convexity being the parameter, for split graphs with convexity on K (resp. I), we show that HCYCLE and HPATH are NP-complete on star-convex (resp. comb) split graphs with convexity on K (resp. I). Further, we show that HCYCLE is NP-complete on \(k_{1,r}\)-free star-convex split graphs with convexity on I, \(r\ge 6\). On the positive side, we show that for \(K_{1,5}\)-free star-convex split graphs with convexity on I, HCYCLE is polynomial-time solvable. Thus, we establish a dichotomy for HCYCLE on star-convex split graphs with convexity on I. We further show that the dominating set problem (DS) and its variants (resp. Connected, Total, Outer-Connected, and Dominating biclique) are NP-complete on star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6). On the parameterized complexity front, we prove that the parameterized version of the domination problem and its variants, with the parameter being the solution size, is not fixed-parameter tractable for star-convex bipartite graphs with diameter 3 (resp. diameter 5, and diameter 6), whereas it is fixed-parameter tractable when the parameter is the number of leaves in the associated star. Further, we show that for star-convex bipartite graphs with diameters 5, and 6, the domination problem and its variants cannot be approximated within \((1-\epsilon )\) ln n unless NP \(\subseteq TIME(2^{n^ {o(1)}})\).