## 1 Introduction

The graphs with forbidden subgraphs possess a nice structural characterization. The structural characterization of these graphs has attracted researchers from both mathematics and computing. The popular graphs are chordal graphs[1], which forbid induced cycles of length at least four, and chordal bipartite graphs[2], which are bipartite graphs that forbid induced cycles of length at least six. Further, these graph classes have a nice structural characterization with respect to minimal vertex separators and a special ordering, namely, perfect elimination ordering[3] among its vertices (edges). These graphs are largely studied in the literature to understand the computational complexity of classical optimization problems such as vertex cover, dominating set, coloring, etc., as these problems are known to be NP-complete in general graphs. Thus, these graphs help to identify the gap between NP-completeness and polynomial-time solvable instances of a combinatorial problem.

The Hamiltonian cycle (path) problem is a famous problem which asks for the presence of a cycle (path) that visits each node exactly once in a graph. Hamiltonian problems play a significant role in various research areas such as circuit design[4], operational research[5], biology[6], etc. On the complexity front, this problem is well-studied and it remains NP-complete on general graphs. Interestingly, this problem is polynomial-time solvable on special graphs such as cographs and permutation graphs [7]. Surprisingly, this problem is NP-complete in chordal graphs[8], bipartite graphs[9] and free graphs.

Haiko Muller[10] has shown that the hamiltonian cycle problem is NP-complete in chordal bipartite graphs by a polynomial-time reduction from the satisfiablity problem . The microscopic view of reduction instances reveals that the instances are -free chordal bipartite graphs. It is natural to study the complexity of hamiltonian cycle problem in -free chordal bipartite graphs and its subclasses. Since -free chordal bipartite graphs are complete bipartite graphs, the first non-trivial graph class in this line of research is -free chordal bipartite graphs. It is known from the literature that problems such as hamiltonian cycle, clique, clique cover, domination, etc., have polynomial-time algorithms in -free graphs. In recent times, the class -free graphs have received a good attention, and problems such as independent set[11] and 3-colourbility[12] have polynomial-time algorithms restricted to -free graphs.

Our work: In this paper, we study the structure of -free chordal bipartite graphs and present a new ordering referred to as Nested Neighbourhood Ordering among its vertices. We present a polynomial-time algorithm for hamiltonian cycle (path) problem using the Nested Neighbourhood Ordering. Further, using this ordering, we present polynomial-time algorithms for longest path, minimum leaf spanning tree and Steiner path problems.

### 1.1 Preliminaries

All the graphs used in this paper are simple, connected and unweighted. For a graph , let denote the vertex set and denote the edge set. The notation represents an edge incident on the vertices and . The degree of a vertex is denoted as , where denotes the set of vertices that are adjacent to . If is disconnected, then denotes the number of connected components in (each component being maximal). A bipartite graph is chordal bipartite if every cycle of length six has a chord. A maximal biclique is a complete bipartite graph such that there is no strict supergraphs or . A maximum biclique is a maximal biclique with the property that is minimum. Note that is an induced path of length and denotes a path that starts at and ends at , and denotes its length. We use and interchangeably.

## 2 Structural Results

In this section, we shall present a structural characterization of -free chordal bipartite graphs. Also, we introduce Nested Neighborhood Ordering among its vertices. We shall fix the following notation to present our results. For a chordal bipartite with bipartition , let and . . We write and , , and , , such that denotes the maximum biclique.

###### Lemma 1

Let be a -free chordal bipartite graph. Then, such that and such that .

###### Proof

Suppose there exists in such that for all in , . Then, is the maximum clique, contradicting the fact that is maximum. Similar argument is true for . Therefore, the lemma. ∎

###### Lemma 2

Let be a -free chordal bipartite graph. Then, , and , .

###### Proof

On the contrary, , . Case 1: . Then, is the maximum biclique, a contradiction. Case 2: . This implies that there exists such that . Since is connected, , say . In , is an induced . Note that, due to the maximality of , as per Lemma 1, we find . This contradicts that is -free. Case 3: and . I.e., such that and , . In , ), , is an induced . Note that the existence of is due to Lemma 1. This is contradicting the -freeness of . Similarly, , can be proved. ∎

###### Lemma 3

Let be a -free chordal bipartite graph. For , , if , then . Similarly, for , if , .

###### Proof

Let us assume to the contrary that . I.e., .

Case 1: . I.e., such that . Since , such that and . Since , vertex is adjacent to at least one more vertex such that . The path ) is an induced . This is a contradiction.

Case 2: , and . I.e., B such that , and , . Since is a connected graph, ), an induced path of length at least 3. The path ) has an induced , . This is a contradiction. Similarly, for all pairs of distinctive vertices with , can be proved. ∎

###### Theorem 2.1

Let be a -free chordal bipartite graphs with being the maximum biclique. Let and are orderings of vertices. If , then . Further, if , then .

###### Proof

We refer to the above ordering of vertices as Nested Neighbourhood Ordering (NNO) of . From now on, we shall arrange the vertices in in non-decreasing order of their degrees so that we can work with NNO of .

## 3 Hamiltonicity in -free Chordal Bipartite graphs

In this section, we shall present polynomial-time algorithms for hamiltonian cycle and path problems in -free chordal bipartite graphs. For a connected graph and set , denotes the number of connected components in the graph induced on the set . It is well-known, due to, Chvatal [13] that if a graph has a hamiltonian cycle, then for every .

###### Theorem 3.1

Let be a -free chordal bipartite graph. has a hamiltonian cycle if and only if and has an ordering , such that , and has an ordering , , .

###### Proof

Necessity: On the contrary, such that is the first vertex in the ordering with . That is, for , and . Let . From Theorem 2.1, we know that . This implies that . This is a contradiction to Chvatal’s necessary condition for hamiltonian cycle. Similarly, in , can be proved.

Sufficiency: Let and . Since has an ordering such that , , for clarity purpose, we define as follows; , , that is, is adjacent to at least two vertices and at most vertices , is adjacent to at least three vertices and at most vertices and similarly is adjacent to at least vertices and at most vertices . Observe that, due to the maximality of , any of can be adjacent to at most vertices of . Similarly, in , for all , .

Let and . The vertices in can be ordered as and the vertices in can be ordered as .
Note that and .
Further, and .
Since , it follows that .
In ,

is a hamiltonian cycle.

The following lemma is well known and is due to Chvatal[13].

###### Lemma 4

Chvatal[13] If a graph has a Hamiltonian path, then for every .

###### Theorem 3.2

Let be a -free chordal bipartite graph. has a hamiltonian path if and only if one of the following is true

(i) and has an ordering, , and has an ordering, , .

(ii) and has an ordering,, and has an ordering, , .

###### Proof

Necessity: (i) Assume to the contrary that such that is the first vertex in the ordering such that . Let . From Theorem 2.1, we know that . Note that, as per the ordering of , . This implies that . Clearly, . Thus, we contradict Chvatal’s necessary condition for hamiltonian path.
Similarly has an ordering such that , .

(ii) For , the argument is similar to the above. Suppose such that is the first vertex in the ordering such that . From Theorem 2.1, . Consider the set and . Further, . Note that and . Since , . Clearly, , contradicting Chvatal’s condition for hamiltonian path.

Sufficiency: (i) Let , and . Consider . . Note that and . In ,

is a hamiltonian path.

(ii) Consider and
.
In , is a hamiltonian path. This completes a proof of this claim.

###### Theorem 3.3

Let be a -free chordal bipartite graph. Finding hamiltonian path and cycle in are polynomial-time solvable.

## 4 Longest paths in -free chordal bipartite graphs

For a connected graph , the longest path is an induced path of maximum length in . Since hamiltonian path is a path of maximum length, finding a longest path is trivially solvable if the input instance is an yes instance of hamiltonian path problem. Thus, the longest path problem is a generalization of hamiltonian path problem, and hence the longest path problem is NP-complete if hamiltonian path problem is NP-complete in the graph class under study. On the other hand, it is interesting to investigate the complexity of longest path problem in graphs where the hamiltonian path problem is polynomial-time solvable. Since, hamiltonian path problem in -free chordal bipartite graphs is polynomial-time solvable, in this section, we shall investigate the complexity of the longest path problem in -free chordal bipartite graphs.

Pruning: We shall now prune by removing vertices that will not be part of any longest path in . Without loss of generality, we assume that has no hamiltonian path, and hence, there must exist vertices in () that violate degree conditions mentioned in Theorem 3.2. As part of pruning, we prune such vertices from . Recall that . Let be the first vertex in with . Remove and relabel the vertices of so that the sequence is reduced to . After, say iterations, becomes such that for . Similarly, after iterations, becomes such that for . From now on, when we refer to (), it refers to the modified (). We define a subgraph on the modified and it is induced on the set .

Note that if , then we get a path in and . If , then we get a path in and .

Claim: is a longest path in .

Observe that if , then the subgraph of induced on the set is an yes instance of the hamiltonian path problem. Similarly, if , then the subgraph of induced on the set is an yes instance of the hamiltonian path problem. Since respects NNO, for all the pruned vertices of , their neighborhood is a subset of . This shows that the pruned vertices of cannot be augmented to to get a longer path in . This proves that is a longest path in .

We define a subgraph on the modified and it is induced on the set .

Note that if , then we get a path in and . If , then we get a path in and .

Claim: is a longest path in .

Observe that if , then the subgraph of induced on the set is an yes instance of the hamiltonian path problem. Similarly, if , then the subgraph of induced on the set is an yes instance of the hamiltonian path problem. Since respects NNO, for all the pruned vertices of , their neighborhood is a subset of . This shows that the pruned vertices of cannot be augmented to to get a longer path in . This proves that is a longest path in .

We define a subgraph induced on the set . Since the subgraph induced on is a complete bipartite graph, is also a complete bipartite graph. Let and . Let be a longest path in . We shall construct a longest path in using longest paths and . We use and interchangeably to refer to , similarly, for and as well.

Case 1: . Assume and .

Case 1.1: . Then, .

Case 1.2: . Then, .

Case 2: . Note that .

Case 2.1: . Then, .

Case 2.2: . Then, , where is the vertex before in the ordering .

Case 3: . Note that and is the vertex before in the ordering .

Case 3.1: . Then, .

Case 3.2: . Then, .

Claim: is a longest path in . Further, can be computed in polynomial time.

Since and are longest paths as per above claims, is a longest path in . As our proofs are constructive in nature, we obtain in polynomial time.

Remark: As an extension of longest path problem, we naturally obtain a minimum leaf spanning tree of , which is a spanning tree of with the minimum number of leaves, in polynomial time. Since respects NNO, the vertices pruned while constructing and , cannot be included as internal vertices of . We shall now construct a minimum leaf spanning tree with as a subtree. (i) the pruned vertices are augmented to as leaves to obtain . (ii) if , then the vertices in not included in are added to as leaves to obtain . Similarly, if , then the vertices in not included in are added to as leaves to obtain . This shows that has leaves.

## 5 Steiner path in -free chordal bipartite graphs

We now generalize the hamiltonian path problem and ask the follwing: given , find a path containing all of minimizing , if exists. Note that, this constrained path problem is the hamiltonian path problem when . This has another motivation as well. For a connected graph , the well-known Steiner tree problem asks for a tree containing all of minimizing . If we ask for a path instead of a tree in the Steiner tree problem, then it is precisely constrained path problem. Due to this reason, we refer to this problem as the Steiner path problem. It is important to highlight that not all input graphs have a solution to the Steiner path problem. We shall present a constructive proof for the existence of Steiner path by case analysis.

Case 1: .

Claim: If is such that for all , , then there exists a Steiner path in . Otherwise, no Steiner path exists in .

Observe that, the vertices in respects degree constraints, due to which they respect NNO. Thus, the graph induced on is an yes instance of the hamiltonian path problem. Therefore, is a Steiner path. Since is an independent set of size , any path containing must have additional vertices and hence is a minimum Steiner path.

Case 2: .

Let . It is easy to see that if , then is a minimum Steiner path. We shall work with .

Claim: If there exists in such that has NNO with of , then there exists a Steiner path. Otherwise, no Steiner path exists in .

Note that the path is a Steiner path of minimum cardinality.

Case 3: and .

Claim: If has NNO with of , then there exists a Steiner path in . Otherwise, no Steiner path exists in .

The path is a minimum Steiner path in .

On the similar line, other cases , , and , and , and and can be proved.

Conclusions and Further Research: In this paper, we have presented structural results on -free chordal bipartite graphs. Subsequently, using these results, we have presented polynomial-time algorithms for hamiltonian cycle, hamiltonian path, longest path and Steiner path problems. These results exploits the nested neighborhood ordering of -free chordal bipartite graphs which is an important contribution of this paper. A natural direction for further research is to study

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