Color-avoiding connected spanning subgraphs with minimum number of edges

TL;DR

This article investigates the problem of determining the maximum number of edges that can be removed from a color-avoiding connected graph so that it remains color- avoidanceing connected, and proves that this problem is NP-hard, then gives a polynomial-time approximation algorithm for it.

Abstract

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 a color-avoiding connected graph so that it remains color-avoiding connected. First, we prove that this problem is NP-hard, 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.

Figures (12)

• Figure 1: An example for a graphic matroid: the ground set of the matroid is $\{ e_1, e_2, e_3, e_4, e_5 \}$ and the non-independent sets are $\{ e_1, e_2, e_3 \}$, $\{ e_3, e_4, e_5 \}$, $\{ e_1, e_2, e_3, e_4 \}$, $\{ e_1, e_2, e_3, e_5 \}$, $\{ e_1, e_2, e_4, e_5 \}$, $\{ e_1, e_3, e_4, e_5 \}$, $\{ e_2, e_3, e_4, e_5 \}$, and $\{ e_1, e_2, e_3, e_4, e_5 \}$.
• Figure 2: An example for an edge-color-avoiding connected graph (left) --- after the removal of edges of any single color, there remains a Hamiltonian path ---, and an example for a not edge-color-avoiding connected graph (right) --- after the removal of the blue (denoted by squares) edges, the bottom right vertex becomes isolated.
• Figure 3: An example for a vertex- and internally vertex-color-avoiding connected graph (left) --- after the removal of vertices of any single color, there remains a Hamiltonian path, thus for any two vertices, there exists a path containing no internal vertices of the removed color ---, and an example for a vertex- but not internally vertex-color-avoiding connected graph (middle) --- there exists no path between the bottom left and the top right vertices which avoids internal vertices of color blue (denoted by squares), thus these two vertices are not internally vertex-blue-avoiding connected, however, they are vertex-blue-avoiding connected since the color of the top right vertex is blue. Finally, an example of a graph which is neither vertex- nor internally vertex-color-avoiding connected (right) --- the red (denoted by triangle) and green (denoted by rhombus) vertices are neither vertex- nor internally vertex-blue-avoiding connected.
• Figure 4: An edge-color-avoiding connected graph on $8$ vertices and with minimum number of edges colored with exactly $4$ colors.
• Figure 5: An edge-color-avoiding connected graph colored with $k=3$ colors and on $n = 7$ vertices, which contains an edge-color-avoiding connected spanning subgraph with $\frac{k \cdot (n-1)}{k-1} = 9$ edges --- for example, such a subgraph is spanned by the green (denoted by rhombi) edges and the lower red (denoted by triangles) and blue (denoted by squares) edges --- and an edge-color-avoiding connected spanning subgraph with $2(n-1) = 12$ edges --- that subgraph is spanned by the red and blue edges. The output of Algorithm \ref{['alg:matr']} can be this latter subgraph, resulting in an approximation ratio of exactly $\frac{2(k-1)}{k}$.

Theorems & Definitions (33)

• Definition 1
• Lemma 2
• Definition 3
• Definition 4
• Definition 5
• Lemma 6
• Theorem 8
• proof
• Corollary 9
• Theorem 10