Induced Graph Packing Problems

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document} be a set of undirected graphs. The induced \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document} -packing problem in an input graph G is to find a subgraph Q of G of maximum size such that each connected component of Q is an induced subgraph of G and is isomorphic to some member of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document} . In this paper we focus on the case when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document} consists of factor-critical graphs and a certain family of ‘propellers’. Clarifying the methods developed in the related theory of non-induced graph packings, we show a Gallai–Edmonds type structure theorem and a Berge–Tutte type minimax formula. We also give an Edmonds type alternating forest algorithm for the case when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document} consists of a sequential set of stars and factor-critical graphs. This simplifies the related result of Egawa, Kano and Kelmans.