We call a (not necessarily properly) edge-colored graph edge-color-avoiding connected if after the removal of edges of any single color, the graph remains connected. For vertex-colored graphs, similar definitions of color-avoiding connectivity can be given. In this article, we investigate the problem of determining the maximum number of edges that can be removed from either an edge- or a vertex-colored, color-avoiding connected graph so that it remains color-avoiding connected. First, we prove that this problem is NP-hard, and then, we give a polynomial-time approximation algorithm for it. To analyze the approximation factor of this algorithm, we determine the minimum number of edges of color-avoiding connected graphs on a given number of vertices and with a given number of colors. Furthermore, we also consider a generalization of edge-color-avoiding connectivity to matroids.

We introduce a new class of inverse optimization problems in which an input solution is given together with k linear weight functions, and the goal is to modify the weights by the same deviation vector p so that the input solution becomes optimal with respect to each of them, while minimizing ‖p‖1. In particular, we concentrate on three problems with multiple weight functions: the inverse shortest s−t path, the inverse bipartite perfect matching, and the inverse arborescence problems. Using LP duality, we give min–max characterizations for the ℓ1-norm of an optimal deviation vector. Furthermore, we show that the optimal p is not necessarily integral even when the weight functions are so, therefore computing an optimal solution is significantly more difficult than for the single-weighted case. We also give a necessary and sufficient condition for the existence of an optimal deviation vector that changes the values only on the elements of the input solution, thus giving a unified understanding of previous results on arborescences and matchings.

Abstract. Let [Formula: see text] denote the maximum number of edges not contained in any monochromatic copy of [Formula: see text] in a [Formula: see text]-coloring of the edges of [Formula: see text], and let [Formula: see text] denote the Turán number of [Formula: see text]. In place of [Formula: see text] we simply write [Formula: see text]. Keevash and Sudakov proved that [Formula: see text] if [Formula: see text] is an edge-critical graph or [Formula: see text] and asked if this equality holds for any graph [Formula: see text]. All known exact values of this question require [Formula: see text] to contain at least one cycle. In this paper we focus on acyclic graphs and present the following results: (1) We prove [Formula: see text] when [Formula: see text] is a spider or a double broom. (2) We show that a tail in [Formula: see text] is a path [Formula: see text] such that [Formula: see text] is only adjacent to [Formula: see text], and [Formula: see text] is only adjacent to [Formula: see text] in [Formula: see text]. We obtain a tight upper bound for [Formula: see text] when [Formula: see text] is a bipartite graph with a tail. This result provides the first bipartite graphs which answer the question of Keevash and Sudakov in the negative. (3) We answer a question of Liu, Pikhurko, and Sharifzadeh who asked if [Formula: see text] when [Formula: see text] is a tree. We provide an upper bound for [Formula: see text] and show it is tight when [Formula: see text] is prime. This provides a negative answer to their question.

A graph is called t-tough if the removal of any vertex set S that disconnects the graph leaves at most |S|∕t components. The toughness of a graph is the largest t for which the graph is t-tough. A graph is minimally t-tough if the toughness of the graph is t and the deletion of any edge from the graph decreases the toughness. The complexity class DP is the set of all languages that can be expressed as the intersection of a language in NP and a language in coNP. In this paper, we prove that recognizing minimally t-tough graphs is DP-complete for any positive rational number t. We introduce a new notion called weighted toughness, which has a key role in our proof.