Parameterized Algorithms for Graph Burning Problem
The Graph Burning problem is defined as follows. At time $$t=0$$ , no vertex of the graph is burned. At each time $$t \ge 1$$ , we choose a vertex to burn. If a vertex is burned at time t, then at time $$t+1$$ each of its unburned neighbors becomes burned. Once a vertex is burned then it remains in...
Saved in:
| Published in | Combinatorial Algorithms Vol. 11638; pp. 304 - 314 |
|---|---|
| Main Authors | , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
2019
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Online Access | Get full text |
| ISBN | 3030250040 9783030250041 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-030-25005-8_25 |
Cover
| Summary: | The Graph Burning problem is defined as follows. At time $$t=0$$ , no vertex of the graph is burned. At each time $$t \ge 1$$ , we choose a vertex to burn. If a vertex is burned at time t, then at time $$t+1$$ each of its unburned neighbors becomes burned. Once a vertex is burned then it remains in that state for all subsequent steps. The process stops when all vertices are burned. The burning number of a graph is the minimum number of steps needed to burn all the vertices of the graph.
Computing the burning number of a graph is $$\mathsf {NP}$$ -complete even on bipartite graphs or trees of maximum degree three. In this paper we study this problem from the parameterized complexity perspective. We show that the problem is fixed-parameter tractable ( $$\mathsf {FPT}$$ ) when parameterized by the distance to cluster or neighborhood diversity. We further study the complexity of the problem on restricted classes of graphs. We show that Graph Burning can be solved in polynomial time on cographs and split graphs. |
|---|---|
| Bibliography: | Original Abstract: The Graph Burning problem is defined as follows. At time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0$$\end{document}, no vertex of the graph is burned. At each time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge 1$$\end{document}, we choose a vertex to burn. If a vertex is burned at time t, then at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t+1$$\end{document} each of its unburned neighbors becomes burned. Once a vertex is burned then it remains in that state for all subsequent steps. The process stops when all vertices are burned. The burning number of a graph is the minimum number of steps needed to burn all the vertices of the graph. Computing the burning number of a graph is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {NP}$$\end{document}-complete even on bipartite graphs or trees of maximum degree three. In this paper we study this problem from the parameterized complexity perspective. We show that the problem is fixed-parameter tractable (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {FPT}$$\end{document}) when parameterized by the distance to cluster or neighborhood diversity. We further study the complexity of the problem on restricted classes of graphs. We show that Graph Burning can be solved in polynomial time on cographs and split graphs. |
| ISBN: | 3030250040 9783030250041 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-030-25005-8_25 |