{'fraction_non_alphanumeric': 0.01488095238095238, 'fraction_numerical': 0.0, 'mean_word_length': 4.341269841269841, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Surviving Rate of Graphs with Bounded Treewidth

The firefighter problem is a discrete-time game on graphs introduced by Hartnell in an attempt to model the spread of fire, diseases, computer viruses and suchlike in a macro-control level. To measure the defence ability of a graph as a whole, Cai and Wang defined the surviving rate of a graph G, which is the average percentage of vertices that can be saved when a fire starts randomly at one vertex of G. In this paper we prove that if k firefighters are available in each round, the surviving rate of graphs with treewidth no more than k approaches one when the number of vertices of the graph tends to infinity.