The complexity of graph contractions

For a fixed pattern graph H, let H-CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. We continue a line of research that was started in 1987 by Brouwer & Veldman, and we determine the computational complexity of H-CONTRACTIBILITY for certain classes of pattern graphs. In particular, we pin-point the complexity for all graphs H with five vertices. Interestingly, in all cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP-complete, the pattern graph H does not have a dominating vertex

On fixed point sets of distinguished collections for groups of parabolic characteristic

We determine the nature of the fixed point sets of groups of order p, acting on complexes of distinguished p-subgroups (those p-subgroups containing p-central elements in their centers). The case when G has parabolic characteristic p is analyzed in detail.Comment: 15 page

Determinants of Contractual Completeness in Franchising

The aim of the study is to explain the determinants of contractual completeness in franchise relationships by formulating and testing various propositions derived from transaction cost theory, agency theory, property rights theory, organizational capability theory and relational view of governance. The degree of contractual completeness depends on behavioural uncertainty (negatively), trust (positively), franchisees’ specific investments (negatively), environmental uncer-tainty (negatively), intangibility of system specific know-how (negatively) and contract design capabilities (positively). The hypotheses are tested with a data base consisting of 52 franchise systems in Austria. The empirical results support the hypotheses regarding behavioural uncertainty, trust and intangible system-specific know-how.franchising;contractual completeness

Contracting to a Longest Path in H-Free Graphs

The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P? and polynomial-time solvable if H = P?. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction

Vertex decompositions of two-dimensional complexes and graphs

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous decomposition techniques for graphs. For example, we prove that it is NP-complete to decide if a graph can be reduced to a discrete graph by a sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug

Contracting to a longest path in H-free graphs

The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction

The Complexity of Contracting Bipartite Graphs into Small Cycles

For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problemtakes as input an undirected simple graph $G$ and determines whether $G$ can betransformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showedthat $C_4$-Contractibility is NP-complete in general graphs. It is easy toverify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski andPaulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ onbipartite graphs for $\ell = 6$ and posed as open problems the status of theproblem when $\ell$ is 4 or 5. In this paper, we show that both$C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartitegraphs.<br