Complexity of C_k-Coloring in Hereditary Classes of Graphs

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs. We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw

Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Let C and D be hereditary graph classes. Consider the following problem: given a graph G in D, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied: - the graphs in C are sparse, i.e., they have linearly many edges in terms of the number of vertices; - the graphs in D admit balanced separators of size governed by their density, e.g., O(Delta) or O(sqrt{m}), where Delta and m denote the maximum degree and the number of edges, respectively; and - the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes C and D: - a largest induced forest in a P_t-free graph can be found in 2^{O~(n^{2/3})} time, for every fixed t; and - a largest induced planar graph in a string graph can be found in 2^{O~(n^{3/4})} time

Completeness for the Complexity Class ∀ ∃ R and Area-Universality

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class ∃R plays a crucial role in the study of geometric problems. Sometimes ∃R is referred to as the ‘real analog’ of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, ∃R deals with existentially quantified real variables. In analogy to Πp2 and Σp2 in the famous polynomial hierarchy, we study the complexity classes ∀∃R and ∃∀R with real variables. Our main interest is the AREA UNIVERSALITY problem, where we are given a plane graph G, and ask if for each assignment of areas to the inner faces of G, there exists a straight-line drawing of G realizing the assigned areas. We conjecture that AREA UNIVERSALITY is ∀∃R -complete and support this conjecture by proving ∃R - and ∀∃R -completeness of two variants of AREA UNIVERSALITY. To this end, we introduce tools to prove ∀∃R -hardness and membership. Finally, we present geometric problems as candidates for ∀∃R -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability

Fixing Improper Colorings of Graphs

International audienceIn this paper we consider a variation of a recoloring problem, called the r-Color-Fixing. Let us have some non-proper r-coloring $\varphi$ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is "the most similar" to $\varphi$, i.e. the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any r \geq 3, but is Fixed Parameter Tractable (FPT), when parametrized by the number of allowed transformations k. We provide an O∗(2^n) algorithm for the problem (for any fixed r) and a linear algorithm for graphs with bounded treewidth. Finally, we investigate the fixing number of a graph G. It is the maximum possible distance (in the number of transformations) between some non-proper coloring of G and a proper one

Parameterized Complexity of Bandwidth of Caterpillars and Weighted Path Emulation

In this paper, we show that Bandwidth is hard for the complexity class W[t] for all t∈ N, even for caterpillars with hair length at most three. As intermediate problem, we introduce the Weighted Path Emulation problem: given a vertex-weighted path PN and integer M, decide if there exists a mapping of the vertices of PN to a path PM, such that adjacent vertices are mapped to adjacent or equal vertices, and such that the total weight of the pre-image of a vertex from PM equals an integer c. We show that Weighted Path Emulation, with c as parameter, is hard for W[t] for all t∈ N, and is strongly NP-complete. We also show that Directed Bandwidth is hard for W[t] for all t∈ N, for directed acyclic graphs whose underlying undirected graph is a caterpillar

Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs.

Let C and D be hereditary graph classes. Consider the following problem: given a graph G∈ D, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C. We prove that it can be solved in 2 o(n) time, where n is the number of vertices of G, if the following conditions are satisfied:the graphs in C are sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs in D admit balanced separators of size governed by their density, e.g., O(Δ) or O(m), where Δ and m denote the maximum degree and the number of edges, respectively; andthe considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes C and D:a largest induced forest in a Pt-free graph can be found in 2O~(n2/3) time, for every fixed t; anda largest induced planar graph in a string graph can be found in 2O~(n2/3) time

Parameterized Complexity of Bandwidth of Caterpillars and Weighted Path Emulation

In this paper, we show that Bandwidth is hard for the complexity class W[t] for all t∈ N, even for caterpillars with hair length at most three. As intermediate problem, we introduce the Weighted Path Emulation problem: given a vertex-weighted path PN and integer M, decide if there exists a mapping of the vertices of PN to a path PM, such that adjacent vertices are mapped to adjacent or equal vertices, and such that the total weight of the pre-image of a vertex from PM equals an integer c. We show that Weighted Path Emulation, with c as parameter, is hard for W[t] for all t∈ N, and is strongly NP-complete. We also show that Directed Bandwidth is hard for W[t] for all t∈ N, for directed acyclic graphs whose underlying undirected graph is a caterpillar