Cograph Research Articles

Exact and parameterized algorithms for the independent cutset problem

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is ▪-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a O⁎(1.4423n)-time algorithm to compute a minimum independent cutset (if any). Since the property of having an independent cutset is MSO1-expressible, our main results are concerned with structural parameterizations for the problem considering parameters incomparable with clique-width. We present ▪-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to P5-free graphs. We close by introducing the notion of α-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.

Dichotomies for Maximum Matching Cut: H-freeness, bounded diameter, bounded radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for (P6+sP2)-free graphs for every s≥0, extending a known result for P5-free graphs (Bouquet and Picouleau, 2020).

Some progress on the Aharoni–Korman conjecture

Aharoni and Korman (1992) [3] conjectured that every ordered set with no infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain.The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for (3+1)-free posets with no infinite antichains.We give a necessary and sufficient condition for a P4-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite P4-free graphs are finite cographs. We also prove that N-free chain complete posets and N-free posets with no infinite antichains are series-parallel.As a consequence, we obtain that the conjecture is true for N-free posets with no infinite antichain.

Hitting all maximum stable sets in P5-free graphs

We prove that every P5-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where Pt denotes the t-vertex path, and for graphs G,H, we say G is H-free if no induced subgraph of G is isomorphic to H).More generally, let us say a class C of graphs is η-bounded if there exists a function h:N→N such that η(G)≤h(ω(G)) for every graph G∈C, where η(G) denotes smallest cardinality of a hitting set of all maximum stable sets in G, and ω(G) is the clique number of G. Also, C is said to be polynomially η-bounded if in addition h can be chosen to be a polynomial.We introduce η-boundedness inspired by a question of Alon (asking how large η(G) can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely,•given a graph G, we have η(H)≤ω(H) for every induced subgraph H of G if and only if G is perfect;•there are graphs G with both η(G) and the girth of G arbitrarily large; and•if C is a hereditary class of graphs which is polynomially η-bounded, then C satisfies the Erdős-Hajnal conjecture. The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all H-free graphs is η-bounded if (and only if) H is a forest. Like χ-boundedness, the case where H is a star is easy to verify, and we prove two non-trivial extensions of this: H-free graphs are η-bounded if (1) H has a vertex incident with all edges of H, or (2) H can be obtained from a star by subdividing at most one edge, exactly once.Unlike χ-boundedness, the case where H is a path is surprisingly hard. Our main result mentioned at the beginning shows that P5-free graphs are η-bounded. The proof is rather involved compared to the classical “Gyárfás path” argument which establishes, for all t, the χ-boundedness of Pt-free graphs. It remains open whether Pt-free graphs are η-bounded for t≥6.It also remains open whether P5-free graphs are polynomially η-bounded, which, if true, would imply the Erdős-Hajnal conjecture for P5-free graphs. But we prove that H-free graphs are polynomially η-bounded if H is a proper induced subgraph of P5. We further generalize the case where H is a 1-regular graph on four vertices, showing that H-free graphs are polynomially η-bounded if H is a forest with no vertex of degree more than one and at most four vertices of degree one.

Some remarks on graphs with no induced subdivision of [formula omitted

For a positive integer n, we use Kn and Pn to denote a complete graph and an induced path on n vertices, respectively. A subdivision of G is a graph obtained from G by replacing the edges of G with independent paths of length at least one between their end vertices. A graph is said to be ISK4-free if it does not contain any subdivision of K4 as an induced subgraph. Lévêque, Maffray and Trotignon conjectured that every ISK4-free graph is 4-colorable. In this paper, we show that this conjecture is true for the class of P6-free graphs.

Can Romeo and Juliet meet? Or rendezvous games with adversaries on graphs

We introduce the rendezvous game with adversaries. In this game, two players, Facilitator and Divider, play against each other on a graph. Facilitator has two agents and Divider has a team of k agents located in some vertices. They take turns in moving their agents to adjacent vertices (or staying put). Facilitator wins if his agents meet in some vertex. Divider aims to prevent the rendezvous of Facilitator's agents. We show that deciding whether Facilitator can win is PSPACE-hard and, when parameterized by k, co-W[2]-hard. Moreover, even deciding whether Facilitator can win within τ steps is co-NP-complete already for τ=2. On the other hand, for chordal and P5-free graphs, we prove that the problem is solvable in polynomial time. Finally, we show that the problem is fixed-parameter tractable parameterized by both the graph's neighborhood diversity and the number of steps τ.

Cops and robber on subclasses of P5-free graphs

The game of cops and robber is a turn-based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say that the team of cops wins the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in a connected graph is called the cop number of the graph. Sivaraman (2019) [16] conjectured that for every t≥5, the cop number of a connected Pt-free graph is at most t−3, where Pt denotes a path on t vertices. Turcotte (2022) [18] showed that the cop number of any 2K2-free graph is at most 2, which was earlier conjectured by Sivaraman and Testa. Note that if a connected graph is 2K2-free, then it is also P5-free. Liu showed that the cop number of a connected (Pt, H)-free graph is at most t−3, where H is a cycle of length at most t or a claw. So the conjecture of Sivaraman is true for (P5, H)-free graphs, where H is a cycle of length at most 5 or a claw. In this paper, we show that the cop number of a connected (P5,H)-free graph is at most 2, where H∈{diamond, paw, K4, 2K1∪K2, K3∪K1}.

On the complexity of matching cut for graphs of bounded radius and H-free graphs

For a connected graph G=(V,E), a matching M⊆E is a matching cut of G if G−M is disconnected. It is known that for an integer d, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most d if d≤2 and NP-complete if d≥3. We prove the same dichotomy for graphs of bounded radius. For a graph H, a graph is H-free if it does not contain H as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for P6-free graphs, extending a recent result of Feghali for P5-free graphs. We then extend our result to hold even for (sP3+P6)-free graphs for every s≥0 and initiate a complexity classification of Matching Cut for H-free graphs.

Binomial edge ideals of cographs

We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.

A note on matching-cut in Pt-free graphs

A matching-cut of a graph is an edge cut that is a matching. The problem matching-cut is that of recognizing graphs with a matching-cut and is NP-complete, even if the graph belongs to one of a number of classes. We initiate the study of matching-cut for graphs without a fixed path as an induced subgraph. We show that matching-cut is in P for P5-free graphs, but that there exists an integer t>0 for which it is NP-complete for Pt-free graphs.

On the chromatic number of some P5-free graphs

Let G be a graph. We say that G is perfectly divisible if for each induced subgraph H of G, V(H) can be partitioned into A and B such that H[A] is perfect and ω(H[B])<ω(H). We use Pt and Ct to denote a path and a cycle on t vertices, respectively. For two disjoint graphs F1 and F2, we use F1∪F2 to denote the graph with vertex set V(F1)∪V(F2) and edge set E(F1)∪E(F2), and use F1+F2 to denote the graph with vertex set V(F1)∪V(F2) and edge set E(F1)∪E(F2)∪{xy|x∈V(F1) and y∈V(F2)}. In this paper, we prove that (i) (P5,C5,K2,3)-free graphs are perfectly divisible, (ii) χ(G)≤2ω2(G)−ω(G)−3 if G is (P5,K2,3)-free with ω(G)≥2, (iii) χ(G)≤32(ω2(G)−ω(G)) if G is (P5,K1+2K2)-free, and (iv) χ(G)≤3ω(G)+11 if G is (P5,K1+(K1∪K3))-free.

The maximum size of an edge 2-neighborhood in P5-free graphs

For a graph G=(V,E) and an edge uv∈E(G), the 2-neighborhood of uv is the set of all edges having at least one endvertex in N(u)∪N(v). A graph is called P5-free if it contains no induced subgraphs isomorphic to a path with 5 vertices. For P5-free graphs, we show that the maximum cardinality of an edge 2-neighborhood is at most 5Δ24, where Δ is the maximum degree of graphs. When Δ is even, this bound is tight and confirms the strong edge coloring conjecture of Erdős and Nešetřil for P5-free graphs.

Efficient computation of the oriented chromatic number of recursively defined digraphs

In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph G=(V,E), an oriented r-coloring for G is a partition of the vertex set V into r independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of G is the smallest integer r such that G permits an oriented r-coloring. Deciding whether an acyclic digraph has an oriented 4-coloring is NP-hard, which motivates to consider the problem on special graph classes.In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 7 for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure.

K-Critical graphs in P5-free graphs

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. Let Pt be the path on t vertices. A graph G is k-vertex-critical if G has chromatic number k but every proper induced subgraph of G has chromatic number less than k. The study of k-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is (k−1)-colorable.In this paper, we initiate a systematic study of the finiteness of k-vertex-critical graphs in subclasses of P5-free graphs. Our main result is a complete classification of the finiteness of k-vertex-critical graphs in the class of (P5,H)-free graphs for all graphs H on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs H using various techniques – such as Ramsey-type arguments and the dual of Dilworth's Theorem – that may be of independent interest.

Toughness and prism-hamiltonicity of [formula omitted]-free graphs

The prism over a graph G is the product G□K2, i.e., the graph obtained by taking two copies of G and adding a perfect matching joining the two copies of each vertex by an edge. The graph G is called prism-hamiltonian if it has a hamiltonian prism. Jung showed that every 1-tough P4-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for k≥1 a P4-free graph has a spanning k-walk (closed walk using each vertex at most k times) if and only if it is 1k-tough. As our main result, we show that for the class of P4-free graphs, the three properties of being prism-hamiltonian, having a spanning 2-walk, and being 12-tough are all equivalent.

Recognition and Optimization Algorithms for P5-Free Graphs

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

On some special classes of contact [formula omitted]-VPG graphs

A graph G is a B0-VPG graph if one can associate a horizontal or vertical path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid-point. A graph G is a contactB0-VPG graph if it is a B0-VPG graph admitting a representation with no one-point paths, no two paths crossing, and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact B0-VPG graphs within four special graph classes: chordal graphs, tree-cographs, P4-tidy graphs and P5-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact B0-VPG graphs.

On efficient domination for some classes of [formula omitted]-free bipartite graphs

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted H-free graph classes such as for 2P3-free chordal graphs while it is solvable in polynomial time for P6-free graphs. Here we focus on bipartite graphs: We show that (weighted) ED can be solved in polynomial time for H-free bipartite graphs when H is P7 or ℓP4 for fixed ℓ, and similarly for P9-free bipartite graphs with vertex degree at most 3, and when H is S2,2,4. Moreover, we show that ED is NP-complete for bipartite graphs with diameter at most 6.

Obstructions for three-coloring graphs without induced paths on six vertices

We prove that there are 24 4-critical P6-free graphs, and give the complete list. We remark that, if H is connected and not a subgraph of P6, there are infinitely many 4-critical H-free graphs. Our result answers questions of Golovach et al. and Seymour.

On efficient domination for some classes of [formula omitted]-free chordal graphs

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted graph classes such as for 2P3-free chordal graphs while it is solvable in polynomial time for P6-free chordal graphs (and even for P6-free graphs). A standard reduction from the NP-complete Exact Cover problem shows that ED is NP-complete for a very special subclass of chordal graphs generalizing split graphs. The reduction implies that ED is NP-complete e.g. for double-gem-free chordal graphs while it is solvable in linear time for gem-free chordal graphs (by various reasons such as bounded clique-width, distance-hereditary graphs, chordal square etc.), and ED is NP-complete for butterfly-free chordal graphs while it is solvable in linear time for 2P2-free graphs.We show that (weighted) ED can be solved in polynomial time for H-free chordal graphs when H is net, extended gem, or S1,2,3.