Volume 24, No 3, 2017, P. 61-79
D. B. Mokeev
On König graphs with respect to $P_4$
We describe the class of graphs whose every induced subgraph has the property: The maximum number of disjoint induced 4-paths is equal to the minimum size of the set of the vertices such that each 4-path contains at least one of them. The description is based on the operation of replacing vertices by cographs which is to the vertices of the graphs obtained from bipartite graphs by subdividing their cycle edges.
Keywords: packing of subgraphs, vertex covering of subgraphs, 4-path, König graph.
Dmitry B. Mokeev 1,2
1. Lobachevsky Nizhny Novgorod State University,
23 Gagarin Ave., 603950 Nizhny Novgorod, Russia
2. Higher School of Economics in Nizhny Novgorod,
25/12 Bolshaya Pechyorskaya St., 603155 Nizhny Novgorod, Russia
Received 30 December 2016
Revised 24 January 2017
 V. E. Alekseev and D. B. Mokeev, König graphs with respect to 3-paths, Diskretn. Anal. Issled. Oper., 19, No. 4, 3–14, 2012.
 D. S. Malyshev, The impact of the growth rate of the packing number of graphs on the computational complexity of the independent set problem, Diskretn. Mat., 25, No. 2, 63–67, 2013. Translated in Discrete Math. Appl., 23, No. 3–4, 245–249, 2013.
 D. S. Malyshev, Classes of graphs critical for the edge list-ranking problem, Diskretn. Anal. Issled. Oper., 20, No. 6, 59–76, 2013. Translated in J. Appl. Ind. Math., 8, No. 2, 245–255, 2014.
 V. E. Alekseev, On easy and hard hereditary classes of graphs with respect to the independent set problem, Discrete Appl. Math., 132, No. 1–3, 17–26, 2004.
 V. E. Alekseev and D. B. Mokeev, König graphs for 3-paths and 3-cycles, Discrete Appl. Math., 204, 1–5, 2016.
 D. G. Corneil, H. Lerchs, and L. Stewart Burlingham, Complement reducible graphs, Discrete Appl. Math., 3, No. 3, 163–174, 1981.
 G. Ding, Z. Xu, and W. Zang, Packing cycles in graphs, II, J. Comb. Theory, Ser. B, 87, No. 2, 244–253, 2003.
 J. Edmonds, Paths, trees, and flowers, Can. J. Math., 17, No. 3–4, 449–467, 1965.
 P. Hell, Graph packings, Electron. Notes Discrete Math., 5, 170–173, 2000.
 D. G. Kirkpatrick and P. Hell, On the completeness of a generalized matching problem, in Proc. X Annu. ACM Symp. Theory Comput., San Diego, USA, May 1–3, 1978, pp. 240–245, ACM, New York, 1978.
 N. V. R. Mahadev and U. N. Peled, Threshold graphs and related topics, North Holland, Amsterdam, 1995 (Ann. Discrete Math., Vol. 56).
 D. B. Mokeev, König graphs for 4-paths, in Models, Algorithms and Technologies for Network Analysis (Proc. 3th Int. Conf. Netw. Anal., Nizhny Novgorod, Russia, May 20–22, 2013), pp. 93–103, Springer, Cham, 2014 (Springer Proc. Math. Stat., Vol. 104).
 R. Yuster, Combinatorial and computational aspects of graph packing and graph decomposition, Comput. Sci. Rev., 1, No. 1, 12–26, 2007.