{'fraction_non_alphanumeric': 0.05026455026455026, 'fraction_numerical': 0.03439153439153439, 'mean_word_length': 3.9802631578947367, '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}}
Connected subgraphs with small degree sums in 3-connected planar graphs

It is well-known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3-connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3-connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that ΣxeV(H) degG(x) ≤ 8t - 1. As a tool for proving this result, we consider decompositions of 3-connected planar graphs into connected subgraphs of order at least t and at most 2t - 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191203, 1999