Hamiltonian path problem is a well studied problem of finding a spanning path in a connected graph. Hamiltonian path problem has substantial scientific attention in modelling various real life problems, and finds applications in physical sciences, operational research [1, 2], etc. This problem has been studied in various perspectives. In the initial stages of study, researchers explored the problem on structural perspective. That is, necessary conditions and sufficient conditions for the existence of Hamiltonian paths in connected graphs. Further, special graphs with bounded graph parameters such as degree, toughness, connectivity, independence number, etc., have been explored for obtaining Hamiltonian paths . Another interesting view on the Hamiltonian problems have been obtained on graphs with forbidden sub graph structures. For example, Hamiltonian paths in claw-free graphs and its sub classes have been explored . Variants of Hamiltonian problems such as Hamiltonian path starting from a specific vertex, Hamiltonian path between a fixed pair of vertices, Hamiltonian connectedness, pancyclicity, etc., have also been explored in the literature. A detailed survey has been compiled by Broersma and Gould [3, 5, 4].
On algorithmic perspective, the problem is NP-complete in general graphs, and in particular, special graph classes such as chordal , bipartite, chordal bipartite , planar , grid graphs , etc. On the other hand, polynomial-time results for the problem have been obtained for interval [10, 11], circular arc [12, 13], proper interval [14, 15], distance hereditary, cocomparability graphs , complete multipartite graphs , etc. It is important to note that although polynomial-time results are known for special graph classes, we still have a “thick complexity line” separating NP-complete instances and polynomial-time solvable instances. For instance, Hamiltonian path problem in chordal graphs is NP-complete and a maximal graph class which is a subclass of chordal graph for which a polynomial-time algorithm is known is the class of interval graphs. However, the complexity line separating chordal graphs (NP-complete instance) and interval graphs (polynomial-time instance) for Hamiltonian path problem is thick. It is important to highlight that there are infinitely many non-interval chordal graphs, for example, chordal graphs with asteroidal triple as a sub graph, on which the complexity of Hamiltonian path is open. To make this line thin, one must do a micro level analysis of the NP-complete reduction of the Hamiltonian path problem in chordal graphs. Further, this asks for a deeper study of the structure of chordal graphs.
In this paper we revisit the Hamiltonian path problem in chordal graphs and present a tight hardness result. We attempt a micro level structural study for Hamiltonian path problem in split graphs and establish that Hamiltonian path problem in -free split graph is NP-complete, which is a popular sub class of chordal graphs. Further, to make the borderline thin between NP-complete instances and polynomial-time instances, we do a deeper investigation of the structure of -free split graphs, which is a major contribution of this paper. To the best of our knowledge, this line of investigation has not been looked at in the literature. The only known results in this context are the study of Hamiltonian cycle in -free and -free split graphs , and the study of Steiner tree in -free and -free split graphs . As a result of our deep structural study, we show that Hamiltonian path problem is polynomial-time solvable in -free split graphs. This brings an interesting dichotomy for Hamiltonian path problem in split graphs.
The rest of the paper is organized as follows. We next present the graph preliminaries. In Section 2 we present the polynomial-time results of the dichotomy. The hardness result is presented in Section 3. The concluding remarks and future work are discussed in Section 4.
We use standard basic graph-theoretic notations. Further, we follow . All the graphs we mention are simple, and unweighted. Graph has vertex set and edge set which we denote using , respectively, once the context is unambiguous. For independent set, maximal clique, and maximum clique we use the standard definitions. Split graphs are -free graphs and the vertex set of a split graph can be partitioned into a clique and an independent set . Such a split graph is denoted as . For a split graph , we assume to be a maximum clique. For , . If , is also denoted as . For a split graph and we define . Accordingly, if , . and . For , represents the subgraph of induced on the vertex set . represents the number of components in graph . For a cycle or a path , by , we mean the visit of vertices in order . Similarly, by , we mean the visit of vertices in order . represents the ordered vertices from to in . For a path of length , for simplicity, we use to denote the underlying set and are end vertices of .
2 Hamiltonian path problem in split graphs : polynomial-time results
We organize our results on Hamiltonian path as Hamiltonian path in -free split graphs and Hamiltonian path in -free split graphs. We present our results on -free split graphs in a systematic way. That is, we shall present Hamiltonian path in -free split graph with , followed by . We make use of the following results from the literature to present our results.
Lemma 1 ()
For a -free split graph , if , then .
Lemma 2 ()
Let be a -free split graph. contains a Hamiltonian cycle if and only if is 2-connected.
Theorem 2.1 ()
Let be a -connected, -free split graph with . Then has a Hamiltonian cycle if and only if there are no short cycles in .
Lemma 3 (Chvatal)
Let be a connected graph. If has a Hamiltonian path, then for every , .
Lemma 4 ()
For a connected split graph with , let , and . If is -free, then .
Corollary 1 (of Lemma 4)
Let be a connected -free split graph with . For every vertex , .
2.1 Results on -free split graphs
Let be a connected -free split graph. contains a Hamiltonian path if and only if has at most vertices such that , and .
If there exists at least three vertices such that , then clearly has no Hamiltonian path.
For the sufficiency, we see the following cases.
Case 1: For every , if , then is 2-connected, and by Lemma 2, has a Hamiltonian cycle. Thus has a Hamiltonian path.
Case 2: If there exists only one vertex with , then observe that is 2-connected. By Lemma 2, there is a Hamiltonian cycle in , which can be easily extended to a Hamiltonian path in .
Case 3: There exists two vertices with . If , it is easy to see that there is a -Hamiltonian path in . If , then we claim that and . Suppose , then let , . Clearly, all the vertices are adjacent to , otherwise induces a . It follows that is a clique of larger size, contradicting the maximality of . Similar arguments hold with respect to the vertex , and hence . Thus we conclude that . From Lemma 1, if , since is connected, . Now we produce a Hamiltonian path in with as follows. Let , such that for all , , . Let be any two elements in . Since , note that for all , . Let , , , and , then is a Hamiltonian path in . can also be written as . This completes a proof of Theorem 2.2.
2.2 Results on -free split graphs
Let be a connected -free split graph with . contains a Hamiltonian path if and only if there exists at most vertices such that , and .
The proof is similar to the proof of Case 3 in Theorem 2.2.
We shall define some special paths and cycles in a -free split graph . We define the restricted bipartite subgraph of as follows. , , and . An induced cycle in is referred to as short cycle in (as well as ) if . An - path is a maximal path in that starts and ends in . Similarly - path and - path are maximal paths in with end vertices in and end vertices in , , respectively. A maximal - path in is referred to as Short - path if . An example is illustrated in Figure 1.
Let be a connected -free split graph with and be the restricted bipartite subgraph of . contains a Hamiltonian path if and only if the following holds true.
1. has no short - path.
2. The number of - paths in is at most 2.
If there exists a short - path in , then note that where , and there is no Hamiltonian path in as per Lemma 3. It is easy to see that if the number of - paths in is more than 2, then there is no spanning path in that includes all the vertices in all such - paths. For sufficiency, we see the following. Since is the restricted bipartite subgraph of , is a collection of maximal paths and short cycles. Moreover, any short cycle in is also a maximal - path in . We initialize a set with the set of maximal paths in . It follows that has at most two - paths. Let and . We now outline a procedure to update in two stages, using which we construct a Hamiltonian path in . In the first stage, for every vertex , which is by definition , include in . Since , observe that any vertex is not adjacent to any internal vertex of paths in . Thus such a vertex is adjacent to the end vertices of paths in . In particular, may be adjacent to some of the newly added in during the first stage. Further, implies that is adjacent to the end vertices of at least two different paths . As a part of the second stage, we update as follows. For every vertex , we find paths such that one of is either a - path or . The paths are replaced with the path in . Let be the resultant set of paths after completing the second stage. If there exists two - paths, then let it be , and if there exists only one - path, then let it be . Then is a Hamiltonian path in . This completes the sufficiency part and a proof of the theorem.
Definition: A connected -free split graph satisfies Property A if , has no short - path, and the sum of the number of - paths and the number of short cycles is at most 2. In a -free split graph with , we define .
Consider a -free split graph with . We shall now show that the number of short cycles in is at most 1 and the length of short cycle is at most 8. Subsequently, if satisfies Property A, then we produce a Hamiltonian path in . Towards this attempt, we bring in a transformation which will transform an instance of into instance . Our results are deep and investigates the structure of the restricted bipartite subgraph of to obtain a Hamiltonian path in .
Let be a connected -free split graph with . Then, the number of short cycles in is at most one. Further, if has a short cycle , then .
For a contradiction assume that there are at least two short cycles in . Let be any two short cycles in . Since , there exists . Clearly, there exists a vertex such that is adjacent to all the vertices in and . It follows that all the vertices in are adjacent to . Note that is a clique of larger size, which contradicts the maximality of clique . For the second part, assume for a contradiction that there exists a short cycle . Consider the cycle such that , , , , . Since , there exists . To complete our proof, we identify a vertex as follows. If , then from Corollary 1, there exists a vertex such that . If , then without loss of generality, we assume . There exists such that . We claim that the vertices are adjacent to , otherwise , induces a . Further , otherwise , induces a . Also , otherwise , induces a . From Corollary 1, it follows that all the vertices in are adjacent to . Suppose there exists such that , then for any , induces a , a contradiction. Finally, is a larger clique, contradicting to the maximality of . Therefore, no such exists. This completes a proof of the lemma.
Definition: Let be a connected -free graph with satisfying property A and be the restricted bipartite subgraph of . By the constructive proof of Theorem 2.4, there exists a collection of vertex disjoint paths containing all the vertices of .
Such a collection is termed as a path collection of .
Let be a -free split graph with , satisfying Property A. For a vertex let . Let , be the restricted bipartite subgraphs of , , respectively and be a path collection of . Clearly, is a subgraph of . Let , be the set containing all the maximal paths of length in . Thus, , where is the set of maximal paths of size where for every , there does not exist such that . Note that has - paths (even length paths) and -
paths (odd length paths). A- path is defined on the vertex set , such that , . We denote such a path as . Similarly, represents an - path with , , and .
Let be a connected -free split graph with , satisfying Property A. If has a short cycle , then there exists a Hamiltonian path in .
Let , . Recall that is the restricted bipartite subgraph of and is a path collection of . From Lemma 5, there exists exactly one short cycle in . Let length of be and is such that . We see the following cases depending on the presence of in .
Case 1: . Consider a vertex . From Corollary 1, . Thus there exists such that . Now we claim that there exists such that . Suppose for a contradiction assume for every , . It follows from Corollary 1 that all the vertices in are adjacent to and is a larger clique, contradicting the maximality of . Thus . Further, from Corollary 1, either or . Without loss of generality, let . Using the above vertices , we claim that in the collection of , for , . Note that . Suppose there exists a path , , then there exists a vertex such that in , . From Corollary 1, . If , then induces a , otherwise induces a . If , then since satisfies Property A, there does not exist such that . It follows that . This is true because has already one short cycle and apart from that it can have at most one - path as per Property A. Since , no such exists. Let and . We construct a path . As per the premise, . Thus , and . It follows that is non-empty, further . From Corollary 1, all the vertices in are adjacent to both and . Suppose there exists such that or , then either or induces a . Thus for every , and . Observe that is a Hamiltonian path in . If , then we see the following. If , since satisfies Property A, i.e., , , . Further, there exists such that . Now we claim that . Suppose not, then there exists such that induces a . Similar to the argument with respect to the vertex , for the vertex , we argue that , and . Let . Then we construct a path . Now we obtain as a Hamiltonian path in . If , then a is a Hamiltonian path in .
Case 2: . Let . Then note that there exists such that . Clearly, for all , . Since is a maximal clique, it follows that there exists such that . We now claim that , . Suppose there exists a path , , then there exists a vertex such that in , . We already observed that . Further, induces a , a contradiction. Thus such a path does not exist. If , then let , . Then is a Hamiltonian path in . If , then is a Hamiltonian path in . This completes the case analysis and a proof of the lemma.
We work on a -free split graph with , satisfying Property A, with , and as defined previously. If has a short cycle, then by Lemma 6, has a Hamiltonian path. If has no short cycles, then note that there exists at most 2 - paths in . For the following claims, we shall consider such a with no short cycle. The structural study of paths in is deep, which is the highlight of this chapter. Now we shall present some claims to show the structural observations of paths in .
If there exists a path such that , then there exists such that .
Assume for a contradiction there exists a path . Let , . From Claim 1 there exists such that . We now claim that . Suppose not, then from Corollary 1, or . Further, , otherwise or has an induced . Similarly, . It follows that has an induced , a contradiction. Thus . If is an odd path, then similar arguments with respect to holds good for the vertex , and hence . Since the clique is maximum in , there exists such that . Further, there exists at least three vertices in adjacent to , otherwise, for some , induces a . Finally, from Corollary 1, either or . It follows that induces a , a contradiction. Thus such a path does not exist. This completes a proof of the claim.
Let , and , be arbitrary paths in . Then there exists such that , , and , .
(of Claim 3) If , , and , are arbitrary paths in , then there exists such that , , , and , .
If there exists , then there does not exist such that .
Assume for a contradiction that there exists such a path . Let , and , . From Claim 3, and . Now we claim . Otherwise, by Corollary 1, or is in . Observe that , otherwise or induces a . Further, either or , otherwise induces a . Now induces a , a contradiction and thus . If is an odd path, then using similar argument, we establish . Now we claim that . Otherwise, by Corollary 1, or