This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. (2024). A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph is said to be strongly conflict-free vertex-connection k -colorable if it admits a (proper) vertex k -coloring such that any two distinct vertices are connected by a conflict-free shortest path. Among others, we show that deciding, for a given graph G and an integer k , whether G is strongly conflict-free vertex-connection k -colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption NP ⊈ coNP/poly , deciding, for a given graph G , whether G is strongly conflict-free vertex-connection 3-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal k - coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number.

Abstract The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are -complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be -complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also -complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d -Cut, for every fixed $$d\ge 1$$ d ≥ 1 , is -complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are -complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for $$\mathcal{H}$$ H -subgraph-free graphs where $$\mathcal{H}$$ H is any finite set of graphs.

International audience

Abstract. For a positive integer [Formula: see text], the [Formula: see text]-recoloring graph of a graph [Formula: see text] has as vertex set all proper [Formula: see text]-colorings of [Formula: see text] with two [Formula: see text]-colorings being adjacent if they differ by the color of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the 7-recoloring graphs of planar graphs, the 5-recoloring graphs of triangle-free planar graphs and the 4-recoloring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose 6-recoloring graph is disconnected, triangle-free planar graphs whose 4-recoloring graph is disconnected, and planar graphs of any given girth whose 3-recoloring graph is disconnected. The main result of this paper consists in showing, via a novel application of the discharging method, that the 4-recoloring graph of every planar graph of girth five is connected. This completes the classification of the connectedness of the recoloring graph for planar graphs of given girth. We also prove some theorems regarding the diameter of the recoloring graph of planar graphs.

Abstract. For an integer [Formula: see text] and a graph [Formula: see text], let [Formula: see text] be the graph that has vertex set all proper [Formula: see text]-colorings of [Formula: see text], and an edge between two vertices [Formula: see text] and [Formula: see text] whenever the coloring [Formula: see text] can be obtained from [Formula: see text] by a single Kempe change. A theorem of Meyniel from 1978 states that [Formula: see text] is connected with diameter [Formula: see text] for every planar graph [Formula: see text]. We significantly strengthen this result by showing that there is a positive constant [Formula: see text] such that [Formula: see text] has diameter [Formula: see text] for every planar graph [Formula: see text].

Abstract A graph isH-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problemsFeedback Vertex SetandOdd Cycle Transversalby showing that they can be solved in polynomial time on $$(sP_1+ P_3)$$ (sP1+P3) -free graphs for every integer $$s\ge 1$$ s≥1 . We show the same result for the variantsConnected Feedback Vertex SetandConnected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known forFeedback Vertex SetandOdd Cycle Transversal. We complement these results by proving thatOdd Cycle TransversalandConnected Odd Cycle Transversalare -complete on $$(P_2+ P_5,P_6)$$ (P2+P5,P6) -free graphs.

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $$k\ge 0$$ , to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for $$P_4$$ -free graphs. We show that it remains polynomial-time solvable for $$P_5$$ -free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer $$k\ge 0$$ . Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems.

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a forest. The set A in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

The NP-complete problem Feedback Vertex Set is to decide if it is possible, for a given integer k>=0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding if a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains in P for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks if a graph has an independent odd cycle transversal of size at most k for a given integer k>=0.

We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.