# Burning Two Worlds

Algorithms for Burning Dense and Tree-Like Graphs
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)

## Abstract

Graph burning is a model for the spread of social influence in networks. The objective is to measure how quickly a fire (e.g., a piece of fake news) can be spread in a network. The burning process takes place in discrete rounds. In each round, a new fire breaks out at a selected vertex and burns it. Meanwhile, the old fires extend to their adjacent vertices and burn them. A burning schedule selects where the new fire breaks out in each round, and the burning problem asks for a schedule that burns all vertices in a minimum number of rounds, termed the burning number of the graph. The burning problem is known to be NP-hard even when the graph is a tree or a disjoint set of paths. For connected graphs, it has been conjectured  that burning takes at most $$\lceil \sqrt{n} \ \rceil$$ rounds.

In this paper, we approach the algorithmic study of graph burning from two directions. First, we consider connected n-vertex graphs with minimum degree $$\delta$$. We present an algorithm that burns any such graph in at most $$\sqrt{\frac{24n}{\delta +1}}$$ rounds. In particular, for graphs with $$\delta \in \varTheta (n)$$, all vertices are burned in a constant number of rounds. More interestingly, even when $$\delta$$ is a constant that is independent of n, our algorithm answers the graph-burning conjecture in the affirmative by burning the graph in at most $$\lceil \sqrt{n} \rceil$$ rounds. Then, we consider burning connected graphs with bounded pathlength or treelength. This includes many graph families, e.g., interval graphs (pathlength 1) and chordal graphs (treelength 1). We show that any connected graph with pathlength pl and diameter d can be burned in $$\lceil \sqrt{d-1} \rceil + pl$$ rounds. Our algorithm ensures an approximation ratio of $$1+o(1)$$ for graphs of bounded pathlength. We also give an algorithm that achieves an approximation ratio of $$2+o(1)$$ for burning connected graphs of bounded treelength. Our approximation factors are better than the best known approximation factor of 3 for burning general graphs.

## Keywords

Graph algorithms Approximation algorithms Graph burning problem Social contagion Pathlength Treelength

## References

1. 1.
Bessy, S., Bonato, A., Janssen, J.C.M., Rautenbach, D., Roshanbin, E.: Burning a graph is hard. Discrete Appl. Math. 232, 73–87 (2017)
2. 2.
Bonato, A., Gunderson, K., Shaw, A.: Burning the plane: densities of the infinite cartesian grid. Preprint (2018)Google Scholar
3. 3.
Bonato, A., Janssen, J.C.M., Roshanbin, E.: Burning a graph as a model of social contagion. In: Workshop of Workshop on Algorithms and Models for the Web Graph, pp. 13–22 (2014)Google Scholar
4. 4.
Bonato, A., Janssen, J.C.M., Roshanbin, E.: How to burn a graph. Internet Math. 12(1–2), 85–100 (2016)
5. 5.
Bonato, A., Kamali, S.: Approximation algorithms for graph burning. In: Theory and Applications of Models of Computation Conference (TAMC), pp. 74–92 (2019)Google Scholar
6. 6.
Bonato, A., Lidbetter, T.: Bounds on the burning numbers of spiders and path-forests. ArXiv e-prints, July 2017Google Scholar
7. 7.
Bond, R.M., et al.: A 61-million-person experiment in social influence and political mobilization. Nature 489(7415), 295–298 (2012)
8. 8.
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)
9. 9.
Das, S., Dev, S.R., Sadhukhan, A., Sahoo, U., Sen, S.: Burning spiders. In: Panda, B.S., Goswami, P.P. (eds.) CALDAM 2018. LNCS, vol. 10743, pp. 155–163. Springer, Cham (2018).
10. 10.
Dourisboure, Y., Gavoille, C.: Tree-decompositions with bags of small diameter. Discrete Math. 307(16), 2008–2029 (2007)
11. 11.
Fajardo, D., Gardner, L.M.: Inferring contagion patterns in social contact networks with limited infection data. Netw. Spat. Econ. 13(4), 399–426 (2013)
12. 12.
Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory Ser. B 16(1), 47–56 (1974)
13. 13.
Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964)
14. 14.
Halin, R.: S-functions for graphs. J. Geom. 8(1–2), 171–186 (1976)
15. 15.
Kamali, S., Miller, A., Zhang, K.: Burning two worlds: Algorithms for burning dense and tree-like graphs. CoRR abs/1909.00530 (2019). http://arxiv.org/abs/1909.00530
16. 16.
Kramer, A.D.I.: The spread of emotion via facebook. In: CHI Conference on Human Factors in Computing Systems, (CHI), pp. 767–770 (2012)Google Scholar
17. 17.
Kramer, A.D.I., Guillory, J.E., Hancock, J.T.: Experimental evidence of massive-scale emotional contagion through social networks. In: Proceedings of the National Academy of Sciences, pp. 8788–8790 (2014)
18. 18.
Land, M.R., Lu, L.: An upper bound on the burning number of graphs. In: Proceedings of Workshop on Algorithms and Models for the Web Graph, pp. 1–8 (2016)Google Scholar
19. 19.
Leitert, A.: Tree-Breadth of Graphs with Variants and Applications. Ph.D. thesis, Kent State University, College of Arts and Sciences, Department of Computer Science (2017)Google Scholar
20. 20.
Liu, H., Zhang, R., Hu, X.: Burning number of theta graphs. Appl. Math. Comput. 361, 246–257 (2019)
21. 21.
Lokshtanov, D.: On the complexity of computing treelength. Discrete Appl. Math. 158(7), 820–827 (2010). third Workshop on GraphClasses, Optimization, and Width Parameters Eugene, Oregon, USA, October 2007
22. 22.
Mitsche, D., Pralat, P., Roshanbin, E.: Burning graphs: a probabilistic perspective. Graphs and Combinatorics 33(2), 449–471 (2017)
23. 23.
Mitsche, D., Pralat, P., Roshanbin, E.: Burning number of graph products. Theor. Comput. Sci. 746, 124–135 (2018)
24. 24.
Robertson, N., Seymour, P.D.: Graph minors iii planar tree-width. J. Comb. Theory Ser. B 36(1), 49–64 (1984)
25. 25.
Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)
26. 26.
Sim, K.A., Tan, T.S., Wong, K.B.: On the burning number of generalized Petersen graphs. Bull. Malays. Math. Sci. Soc. 6, 1–14 (2017) Google Scholar
27. 27.
Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2003).
28. 28.
Šimon, M., Huraj, L., Dirgova Luptáková, I., Pospichal, J.: Heuristics for spreading alarm throughout a network. Appl. Sci. 9(16), 3269 (2019).