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A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours. By extension, a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph $G$ and an integer $p$, the Colourful Components problem asks whether there exist at most $p$ edges whose removal makes $G$ colourful and the Colourful Partition problem asks whether there exists a partition of $G$ into at most $p$ colourful components. In order to refine our understanding of the complexity of the problems on trees, we study both problems on $k$-caterpillars, which are trees with a central path $P$ such that every vertex not in $P$ is within distance $k$ from a vertex in $P$. We prove that Colourful Components and Colourful Partition are NP-complete on $4$-caterpillars with maximum degree $3$, $3$-caterpillars with maximum degree $4$ and $2$-caterpillars with maximum degree $5$. On the other hand, we show that the problems are linear-time solvable on $1$-caterpillars. Hence, our results imply two complexity dichotomies on trees: Colourful Components and Colourful Partition are linear-time solvable on trees with maximum degree $d$ if $d \leq 2$ (that is, on paths), and NP-complete otherwise; Colourful Components and Colourful Partition are linear-time solvable on $k$-caterpillars if $k \leq 1$, and NP-complete otherwise. We leave three open cases which, if solved, would provide a complexity dichotomy for both problems on $k$-caterpillars, for every non-negative integer $k$, with respect to the maximum degree. We also show that Colourful Components is NP-complete on $5$-coloured planar graphs with maximum degree $4$ and on $12$-coloured planar graphs with maximum degree $3$. Our results answer two open questions of Bulteau et al. mentioned in [30th Annual Symposium on Combinatorial Pattern Matching, 2019].

We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of finding a PDS of maximum size is APX -hard on split graphs, and NP -hard on bipartite graphs. We also show that deciding if a PDS is inclusion-wise maximal is co-NP -complete on bipartite graphs. Nevertheless, we present a simple polynomial-time ( 2 − 2 Δ + 1 ) -approximation algorithm for the problem, where Δ is the maximum degree of the graph. Finally, we show that all Hamiltonian cubic graphs with n vertices (except two) have a PDS of size ⌊ 2 n + 1 3 ⌋ , which we prove to be an upper bound on the size of a PDS in cubic graphs.

The Min Anonymous-Edge-Rotation problem asks for an input graph G and a positive integer k to find a minimum number of edge rotations that transform G into a graph such that for each vertex there are at least k − 1 other vertices of the same degree (a k-degree-anonymous graph). In this paper, we prove that the Min Anonymous-Edge-Rotation problem is NP-hard even for k = n / q , where n is the order of a graph and q any positive integer, q ≥ 3 . We argue that under some constrains on the number of edges in a graph and k, Min Anonymous-Edge-Rotation is polynomial-time 2-approximable. Furthermore, we show that the problem is solvable in polynomial time for any graph when k = n and for trees when k = θ ( n ) . Additionally, we establish sufficient conditions for an input graph and k such that a solution for Min Anonymous-Edge-Rotation exists.

The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from $$\frac{123}{122}$$ 123 122 (Karpinski et al. in J Comput Syst Sci 81(8):1665–1677, 2015) to $$\frac{117}{116}$$ 117 116 . This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems.

A proportionally dense subgraph (PDS) is an induced subgraph of a graph such that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study a partition of a graph into two proportionally dense subgraphs, namely a 2-PDS partition, with and without additional constraint of connectivity of the subgraphs. We present two infinite classes of graphs: one with graphs without a 2-PDS partition, and another with graphs that only admit a disconnected 2-PDS partition. These results answer some questions proposed by Bazgan et al. [Algorithmica 80(6)(2018), 1890–1908].

A graph is k-degree-anonymous if for each vertex there are at least \(k-1 \) other vertices of the same degree in the graph. Min Anonymous-Edge-Rotation asks for a given graph G and a positive integer k to find a minimum number of edge rotations that transform G into a k-degree-anonymous graph. In this paper, we establish sufficient conditions for an input graph and k ensuring that a solution for the problem exists. We also prove that the Min Anonymous-Edge-Rotation problem is NP-hard even for \(k=n/3\), where n is the order of a graph. On the positive side, we argue that under some constraints on the number of edges in a graph and k, Min Anonymous-Edge-Rotation is polynomial-time 2-approximable. Moreover, we show that the problem is solvable in polynomial time for any graph when \(k=n\) and for trees when \(k=\theta (n)\).

A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours, and a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph, the Colourful Components problem asks whether there exist at most p edges whose removal makes the graph colourful, and the Colourful Partition problem asks whether there exists a partition of the vertex set with at most p parts such that each part induces a colourful component. We study the problems on k-caterpillars (caterpillars with hairs of length at most k) and explore the boundary between polynomial and NP-complete cases. It is known that the problems are NP-complete on 2-caterpillars with unbounded maximum degree. We prove that both problems remain NP-complete on binary 4-caterpillars and on ternary 3-caterpillars. This answers an open question regarding the complexity of the problems on trees with maximum degree at most 5. On the positive side, we give a linear time algorithm for 1-caterpillars with unbounded degree, even if the backbone is a cycle, which outperforms the previous best complexity for paths and widens the class of graphs. Finally, we answer an open question regarding the complexity of Colourful Components on graphs with maximum degree at most 5. We show that the problem is NP-complete on 5-coloured planar graphs with maximum degree 4, and on 12-coloured planar graphs with maximum degree 3. Since the problem can be solved in polynomial-time on graphs with maximum degree 2, the results are the best possible with regard to the maximum degree.

The investigation of the asymptotic behaviour of various graph parameters in powers of a graph is motivated by problems in information theory and extremal combinatorics. Considering various parameters and/or various notions of graph powers we can arrive at different notions of graph capacities, of which the Shannon capacity is best known. Here we study a related notion of the so-called conjunctive capacity of a graph G, CAND(G), introduced and studied by Gargano, Körner and Vaccaro. To determine CAND(G) is a convex programming problem. We show that the optimal solution to this problem is unique and describe the structure of the solution in any (simple) graph. We prove that its reciprocal value vcC(G):=1/CAND(G) is an optimal solution of the newly introduced problem of Minimum Capacitary Vertex Cover that is closely related to the LP-relaxation of the Minimum Vertex Cover Problem. We also describe its close connection with the binding number/binding set of a graph, and with the strong crown decomposition of graphs.

We consider the complexity of the firefighter problem where a budget of b≥1 firefighters are available at each time step. This problem is known to be NP-complete even on trees of degree at most three and b=1 [S. Finbow, A. King, G. MacGillivray, and R. Rizzi. The firefighter problem for graphs of maximum degree three. Discrete Mathematics, 307(16):2094-2105, 2007] and on trees of bounded degree (b+3 for any fixed b≥2 [C. Bazgan, M. Chopin, and B. Ries. The firefighter problem with more than one firefighter on trees. Discrete Applied Mathematics, 161(7-8):899-908, 2013]. In this paper we provide further insight into the complexity landscape of the problem by showing a complexity dichotomy result with respect to the parameters pathwidth and maximum degree of the input graph. More precisely, first, we prove that the problem is NP-complete even on trees of pathwidth at most three for any b≥1. Then we show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph. Finally, we show that the problem remains NP-complete on very dense graphs, namely co-bipartite graphs, but is fixed-parameter tractable with respect to the parameter "cluster vertex deletion".

Bansal and Sviridenko [N. Bansal, M. Sviridenko, New approximability and inapproximability results for 2-dimensional bin packing, in: Proceedings of the 15th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, 2004, pp. 189–196] proved that there is no asymptotic PTAS for 2-dimensional Orthogonal Bin Packing (without rotations), unless P=NP. We show that similar approximation hardness results hold for several 2- and 3-dimensional rectangle packing and covering problems even if rotations by ninety degrees are allowed. Moreover, for some of these problems we provide explicit lower bounds on asymptotic approximation ratio of any polynomial time approximation algorithm. Our hardness results apply to the most studied case of 2-dimensional problems with unit square bins, and for 3-dimensional strip packing and covering problems with a strip of unit square base.

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree.

The Steiner tree problem on weighted graphs seeks a minimum weight subtree containing a given subset of the vertices (terminals). We show that it is NP-hard to approximate the Steiner tree problem within a factor 96/95. Our inapproximability results are stated in a parametric way, and explicit hardness factors would be improved automatically by providing gadgets and/or expanders with better parameters.

The paper studies crown reductions for the Minimum Weighted Vertex Cover problem introduced recently in the unweighted case by Fellows et al. [Blow-Ups, Win/Win's and crown rules: some new directions in FPT, in: Proceedings of the 29th International Workshop on Graph Theoretic Concepts in Computer Science (WG’03), Lecture notes in computer science, vol. 2880, 2003, pp. 1–12, Kernelization algorithms for the vertex cover problem: theory and experiments, in: Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), New Orleans, Louisiana, January 2004, pp. 62–69]. We describe in detail a close relation of crown reductions to Nemhauser and Trotter reductions that are based on the linear programming relaxation of the problem. We introduce and study the so-called strong crown reductions, suitable for finding (or counting) all minimum vertex covers, or finding a minimum vertex cover under some additional constraints. It is described how crown decompositions and strong crown decompositions suitable for such problems can be computed in polynomial time. For weighted König-Egerváry graphs (G,w) we observe that the set of vertices belonging to all minimum vertex covers, and the set of vertices belonging to no minimum vertex covers, can be efficiently computed.Further, for some specific classes of graphs, simple algorithms for the MIN-VC problem with a constant approximation factor r

In this article we present a structural characterization of graphswithout K_5 and the octahedron as a minor. We introduce semiplanargraphs as arbitrary sums of planar graphs, and give their characteriza-tion in terms of excluded minors. Some other excluded minor theoremsfor 3-connected minors are shown.

We investigate the structural and algorithmic properties of 2-community structure in graphs introduced by Olsen [A general view on computing communities, Mathematical Social Sciences 66(3):331--336, 2013]. A 2-community structure is a partition of vertex set into two parts such that for each vertex of the graph number of neighbours in/outside own part is in correlation with sizes of parts. We show that every 3-regular graph has a 2-community structure which can be found in polynomial time, even if the subgraphs induced by each partition must be connected. We introduce a concept of a 2-weak community and prove that it is NP-complete to find a balanced 2-weak community structure in general graphs evenwith additional request of connectivity for both parts. On the other hand, the problem can be solved in polynomial time in graphs of degree at most 3.

We provide the first interesting explicit lower bounds on efficient approximability for two closely related optimization problems in graphs, MINIMUM EDGE DOMINATING SET and MINIMUM MAXIMAL MATCHING. We show that it is NP-hard to approximate the solution of both problems to within any constant factor smaller than $${\frac{7}{6}}$$ . The result extends with negligible loss to bounded degree graphs and to everywhere dense graphs.

In this paper we theoretically study the MINIMUM RANGE ASSIGNMENT problem in static ad-hoc networks with arbitrary structure, where the transmission distances can violate triangle inequality. We consider two versions of the MINIMUM RANGE ASSIGNMENT problem, where the communication graph has to fulfill either the h-strong connectivity condition (MIN-RANGE(h-SC)) or the h-broadcast condition (MIN-RANGE(h-B)). Both homogeneous and non-homogeneous cases are studied. By approximating arbitrary edge-weighted graphs by paths, we present probabilistic O(logn)-approximation algorithms for MIN-RANGE(h-SC)and MIN-RANGE(h-B), which improves the previous best ratios O(lognloglogn) and O(n^2lognloglogn), respectively [D. Ye, H. Zhang, The range assignment problem in static ad-hoc networks on metric spaces, Proceedings of the 11th Colloquium on Structural Information and Communication Complexity, Sirocco 2004, Lecture Notes in Computer Science, vol. 3104, pp. 291-302]. The result for MIN-RANGE(h-B)matches the lower bound [G. Rossi, The range assignment problem in static ad-hoc wireless networks. Ph.D. Thesis, 2003] for the case that triangle inequality for transmission distance holds (which is a special case of our model). Furthermore, we show that if the network fulfils certain property and the distance power gradient @a is sufficiently small, the approximation ratio is improved to O((loglogn)^@a). Finally we discuss the applications of our algorithms in mobile ad-hoc networks.

The paper is concerned with partial k-trees whose chromatic number is maximal, i.e. equal to (k + 1). We have proved that any such graph contains a triangle (if k ≥ 3), but need not contain a clique on[k+5/2] vertices as a subgraph.

It is known that the class of graphs with treewidth (resp. pathwidth) bounded by a constant w can be characterized by a finite obstruction set obs(TW(w)) (resp. obs(PW(w))). These obstruction sets are known for w⩽3 so far. In this paper we give a structural characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with a fixed number of vertices in terms of subgraphs of the complement. Our approach also essentially simplifies known characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with (w+3) vertices.Also for any w⩾3 a graph from obs(TW(w))⧹obs(PW(w)) is constructed, that solves an open problem.

We consider the complexity of the firefighter problem where a budget of \({b \ge 1}\) firefighters are available at each time step. This problem is proved to be NP-complete even on trees of degree at most three and \({b = 1}\) [10] and on trees of bounded degree \(b+3\) for any fixed \(b \ge 2\) [3]. In this paper, we provide further insight into the complexity landscape of the problem by showing a complexity dichotomy result with respect to the parameters pathwidth and maximum degree of the input graph. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any \(b \ge 1\). Then we show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter “pathwidth” and “maximum degree” of the input graph. Finally, we show that the problem remains NP-complete on very dense graphs, namely co-bipartite graphs, but is fixed-parameter tractable with respect to the parameter “cluster vertex deletion”.

The approximability of the following optimization problem is investigated: Given a connected graph G = (V, E), find the maximally balanced connected partition for G, i.e. a partition (V1, V2) of V into disjoint sets V1 and V2 such that both subgraphs of G induced by V1 and V2 are connected, and maximize an objective function “balance”, B(V1, V2) = min(¦V1¦, ¦V2¦). We prove that for any ϵ > 0 it is NP-hard (even for bipartite graphs) to approximate the maximum balance of the connected partition for G = (V, E) with an absolute error guarantee of ¦V¦1 − e. On the other hand, we give a polynomial-time approximation algorithm that solves the problem within 43 even when vertices of G are weighted. The variation of the problem is equivalent to the Maximally Balanced Spanning Tree Problem studied by Galbiati, Maffioli and Morzenti (1995). Our simple polynomial-time algorithm approximates the solution of that problem within 1.072.

The investigation of the asymptotic behaviour of various graph parameters in powers of a fixed graph G = (V,E) is motivated by problems in information theory and extremal combinatorics. Considering various parameters and/or various notions of graph powers we can arrive at different notions of graph capacities, of which the Shannon capacity is best known. Here we study a related notion of the so-called conjunctive capacity of a graph G, C AND(G), introduced and studied by Gargano, Korner and Vaccaro [5], [6]. To determine C AND(G) is a convex programming problem. In this paper we show that the optimal solution to this problem is unique and describe the structure of the solution in any (simple) graph. We show that its reciprocal value \(vc_C(G):=\frac{1}C_{\text{AND}(G)}\) is an optimal solution of the newly introduced problem of Minimum Capacitary Vertex Cover that is closely related to the LP-relaxation of the Minimum Vertex Cover Problem. We also describe its close connection with the binding number/binding set of a graph, and with the strong crown decomposition of graphs introduced in [2].

The MINIMUM 2SAT-DELETION problem is to delete the minimum number of clauses in a 2SAT instance to make it satisfiable. It is one of the prototypes in the approximability hierarchy of minimization problems Khanna et al. [Constraint satisfaction: the approximability of minimization problems, Proceedings of the 12th Annual IEEE Conference on Computational Complexity, Ulm, Germany, 24–27 June, 1997, pp. 282–296], and its approximability is largely open. We prove a lower approximation bound of View the MathML source8√5-15≈2.88854, improving the previous bound of View the MathML source10√5-21≈1.36067 by Dinur and Safra [The importance of being biased, Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), May 2002, pp. 33–42, also ECCC Report TR01-104, 2001]. For highly restricted instances with exactly four occurrences of every variable we provide a lower bound of View the MathML source 3/2. Both inapproximability results apply to instances with no mixed clauses (the literals in every clause are both either negated, or unnegated).We further prove that any k -approximation algorithm for the MINIMUM 2SAT-DELETION problem polynomially reduces to a (2-2/(k+1))(2-2/(k+1))-approximation algorithm for the MINIMUM VERTEX COVER problem.One ingredient of these improvements is our proof that the MINIMUM VERTEX COVER problem is hardest to approximate on graphs with perfect matching. More precisely, the problem to design a ρρ-approximation algorithm for the MINIMUM VERTEX COVER on general graphs polynomially reduces to the same problem on graphs with perfect matching. This improves also on the results by Chen and Kanj [On approximating minimum vertex cover for graphs with perfect matching, Proceedings of the 11st ISAAC, Taipei, Taiwan, Lecture Notes in Computer Science, vol. 1969, Springer, Berlin, 2000, pp. 132–143].

In this paper we prove that the Precoloring Extension problem on graphs of maximum degree 3 is polynomially solvable, but even its restricted version with 3 colors is NP-complete on planar bipartite graphs of maximum degree 4. The restricted version of List Coloring, in which the union of all lists consists of 3 colors, is shown to be NP-complete on planar 3-regular bipartite graphs.

In this paper we study the Minimum Range Assignment problem in static ad-hoc networks with arbitrary structure, where the transmission distances can violate triangle inequality. We consider two versions of the Minimum Range Assignment problem, where the communication graph has to fulfill either the h-strong connectivity condition (Min-Range(h-SC)) or the h-broadcast condition (Min-Range(h-B)). Both homogeneous and non-homogeneous cases are studied. By approximating arbitrary edge-weighted graphs by paths, we present probabilistic O(log n)-approximation algorithms for Min-Range(h-SC) and Min-Range(h-B), which improves the previous best ratios O(log n loglog n) and O(n2log n loglog n), respectively [21]. The result for Min-Range(h-B) matches the lower bound [20] for the case that triangle inequality for transmission distance holds (which is a special case of our model). Furthermore, we show that if the network fulfils certain property and the distance power gradient α is sufficiently small, the approximation ratio is improved to O((loglog n)α).

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. We state the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. For most of dominating set problems we prove asymptotically almost tight lower bounds. The results are applied to improve the lower bounds for other related problems such as the Maximum Induced Matching problem and the Maximum Leaf Spanning Tree problem.

We obtain slightly improved upper bounds on efficient approximability of the Maximum Independent Set problem in graphs of maximum degree at most B (shortly, B-MaxIS), for small B≥ 3. The degree-three case plays a role of the central problem, as many of the results for the other problems use reductions to it. Our careful analysis of approximation algorithms of Berman and Fujito for 3-MaxIS shows that one can achieve approximation ratio arbitrarily close to \(3-\frac{\sqrt{13}}{2}\). Improvements of an approximation ratio below \(\frac65\) for this case translate to improvements below \(\frac{B+3}{5}\) of approximation factors for B-MaxIS for all odd B. Consequently, for any odd B≥ 3, polynomial time algorithms for B-MaxIS exist with approximation ratios arbitrarily close to \(\frac{B+3}5-\frac{4(5\sqrt{13}-18)}5\frac{(B-2)!!}{(B+1)!!}\). This is currently the best upper bound for B-MaxIS for any odd B, 3≤ B

We improve on the classical Nemhauser-Trotter Theorem, which is a key tool for the Minimum (Weighted) Vertex Cover problem in the design of both, approximation algorithms and exact fixed-parameter algorithms. Namely, we provide in polynomial time for a graph G with vertex weights w : V →〈0, ∞ 〉 a partition of V into three subsets V 0, V 1, \(V_{\frac{1}{2}}\), with no edges between V 0 and \(V_{\frac{1}{2}}\) or within V 0, such that the size of a minimum vertex cover for the graph induced by \(V_{\frac{1}{2}}\) is at least \(\frac{1}{2}w(V_{\frac{1}{2}})\), and every minimum vertex cover C for (G, w) satisfies \(V_1 \subseteq C \subseteq V_1 \cup V_{\frac{1}{2}}\).

The Minimum 2SAT-Deletion problem is to delete the minimum number of clauses in a 2SAT instance to make it satisfiable. It is one of the prototypes in the approximability hierarchy of minimization problems [8], and its approximability is largely open. We prove a lower approximation bound of \(8\sqrt 5-15\approx 2.88854\), improving the previous bound of \(10\sqrt5-21\approx 1.36067\) by Dinur and Safra [5]. For highly restricted instances with exactly 4 occurrences of every variable we provide a lower bound of \(\frac32\). Both inapproximability results apply to instances with no mixed clauses (the literals in every clause are both either negated, or unnegated).

We provide the first interesting explicit lower bounds on efficient approximability for two closely related optimization problems in graphs, MINIMUM EDGE DOMINATING SET and MINIMUM MAXIMAL MATCHING. We show that it is NP-hard to approximate the solution of both problems to within any constant factor smaller than 7/6. The result extends with negligible loss to bounded degree graphs and to everywhere dense graphs.

We study small degree graph problems such as Maximum Independent Set and Minimum Node Cover and improve approximation lower bounds for them and for a number of related problems, like Max- B -Set Packing, Min- B -Set Cover, Max-Matching in B-uniform 2-regular hypergraphs. For example, we prove NP-hardness factor of \(\frac{95}{94}\) for Max-3DM, and factor of \(\frac{48}{47}\) for Max-4DM; in both cases the hardness result applies even to instances with exactly two occurrences of each element.

The paper contributes to the systematic study (started by Berman and Karpinski) of explicit approximability lower bounds for small occurrence optimization problems.We present parametrized reductions for some packing and covering problems, including 3-Dimensional Matching, and prove the best known inapproximability results even for highly restricted versions of them. For example, we show that it is NP-hard to approximate Max-3-DM within 139/138 even on instances with exactly two occurrences of each element. Previous known hardness results for bounded occurence case of the problem required that the bound is at least three, and even then no explicit lower bound was known. New structural results which improve the known bounds for 3-regular amplifiers and hence the inapproximability results for numerous small occurrence problems studied earlier by Berman and Karpinski are also presented.

Steiner tree problem in weighted graphs seeks a minimum weight subtree containing a given subset of the vertices (terminals). We show that it is NP-hard to approximate the Steiner tree problem within 96/95. Our inapproximability results are stated in parametric way and can be further improved just providing gadgets and/or expanders with better parameters. The reduction is from Hastad's inapproximability result for maximum satisfiability of linear equations modulo 2 with three unknowns per equation. This was first used for the Steiner tree problem by Thimm whose approach was the main starting point for our results.

In this paper we deal with the d-PRECOLORING EXTENSION (d-PREXT) problem in various classes of graphs. The d-PREXT problem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain atmost d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it.We present a linear time algorithm for both, the decision and the search version of d-PREXT, in the following cases: (i) restricted to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restrictedto the class of partial k-trees (without any size restriction on S).We also study the following problem related to d-PREXT: given an instance of the d-PREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to usefor precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.

International audience; We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331–336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least |V|−3, trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.