## Abstract

For a given graph *G*, a maximum internal spanning tree of *G* is a spanning tree of *G* with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.

This is a preview of subscription content, access via your institution.

## Data availibility

Enquiries about data availability should be directed to the authors.

## References

Binkele-Raible D, Fernau H, Gaspers S et al (2013) Exact and parameterized algorithms for max internal spanning tree. Algorithmica 65(1):95–128

Chen ZZ, Harada Y, Guo F et al (2018) An approximation algorithm for maximum internal spanning tree. J Comb Optim 35(3):955–979

Cohen N, Fomin FV, Gutin G et al (2010) Algorithm for finding k-vertex out-trees and its application to k-internal out-branching problem. J Comput Syst Sci 76(7):650–662

Fomin FV, Gaspers S, Saurabh S et al (2013) A linear vertex kernel for maximum internal spanning tree. J Comput Syst Sci 79(1):1–6

Garey MR, Johnson DS (1979) Computers and intractability, vol 174. freeman San Francisco

Heggernes P, Kratsch D (2007) Linear-time certifying recognition algorithms and forbidden induced subgraphs. Nord J Comput 14(1–2):87–108

Heggernes P, Van’t Hof P, Lokshtanov D et al (2012) Computing the cutwidth of bipartite permutation graphs in linear time. SIAM J Discret Math 26(3):1008–1021

Jung HA (1978) On a class of posets and the corresponding comparability graphs. J Comb Theory Series B 24(2):125–133

Knauer M, Spoerhase J (2015) Better approximation algorithms for the maximum internal spanning tree problem. Algorithmica 71(4):797–811

Lai TH, Wei SS (1993) The edge hamiltonian path problem is np-complete for bipartite graphs. Inf Process Lett 46(1):21–26

Lai TH, Wei SS (1997) Bipartite permutation graphs with application to the minimum buffer size problem. Discret Appl Math 74(1):33–55

Lerchs H (1972) On the clique-kernel structure of graphs. Dept of Computer Science, University of Toronto 1

Li W, Wang J, Chen J, et al (2015) A 2k-vertex kernel for maximum internal spanning tree. In: Workshop on algorithms and data structures, Springer, pp 495–505

Li W, Cao Y, Chen J et al (2017) Deeper local search for parameterized and approximation algorithms for maximum internal spanning tree. Inf Comput 252:187–200

Li X, Zhu D (2014) Approximating the maximum internal spanning tree problem via a maximum path-cycle cover. In: International symposium on algorithms and computation, Springer, pp 467–478

Li X, Feng H, Jiang H et al (2018) Solving the maximum internal spanning tree problem on interval graphs in polynomial time. Theor Comput Sci 734:32–37

Li X, Zhu D, Wang L (2021) A 4/3-approximation algorithm for the maximum internal spanning tree problem. J Comput Syst Sci 118:131–140

Lin R, Olariu S, Pruesse G (1995) An optimal path cover algorithm for cographs. Comput Math Appl 30(8):75–83

Lu HI, Ravi R (1992) The power of local optimization: Approximation algorithms for maximum-leaf spanning tree. In: Proceedings of the annual allerton conference on communication control and computing, University of Illinois, pp 533–533

Müller H (1996) Hamiltonian circuits in chordal bipartite graphs. Discret Math 156(1–3):291–298

Pak-Ken W (1999) Optimal path cover problem on block graphs. Theore Comput Sci 225(1–2):163–169

Prieto E, Sloper C (2003) Either/or: Using vertex cover structure in designing fpt-algorithms—the case of k-internal spanning tree. In: Workshop on algorithms and data structures, Springer, pp 474–483

Salamon G (2009) Approximating the maximum internal spanning tree problem. Theor Comput Sci 410(50):5273–5284

Salamon G (2010) Degree-based spanning tree optimization. PhD Thesis

Salamon G, Wiener G (2008) On finding spanning trees with few leaves. Inf Process Lett 105(5):164–169

Seinsche D (1974) On a property of the class of n-colorable graphs. J Comb Theory Series B 16(2):191–193

Spinrad J, Brandstädt A, Stewart L (1987) Bipartite permutation graphs. Discret Appl Math 18(3):279–292

Srikant R, Sundaram R, Singh KS et al (1993) Optimal path cover problem on block graphs and bipartite permutation graphs. Theor Comput Sci 115(2):351–357

## Funding

The authors have not disclosed any funding.

## Author information

### Authors and Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

The authors have not disclosed any competing interests.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Michael C. Wigal is supported by an NSF Graduate Research Fellowship under Grant No. DGE-1650044.

## Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

## About this article

### Cite this article

Sharma, G., Pandey, A. & Wigal, M.C. Algorithms for maximum internal spanning tree problem for some graph classes.
*J Comb Optim* **44**, 3419–3445 (2022). https://doi.org/10.1007/s10878-022-00897-4

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10878-022-00897-4

### Keywords

- Maximum internal spanning tree
- Bipartite graphs
- Chordal graphs
- Optimal path cover
- NP-completeness
- Graph Algorithms