On decision and optimization (k,l)-graph sandwich problems

AbstractA graph G is (k,l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k,l)-Graph Sandwich Problem asks, given two graphs G1=(V,E1) and G2=(V,E2), whether there exists a graph G=(V,E) such that E1⊆E⊆E2 and G is (k,l). In this paper, we prove that the (k,l)-Graph Sandwich Problem is NP-complete for the cases k=1 and l=2; k=2 and l=1; or k=l=2. This completely classifies the complexity of the (k,l)-Graph Sandwich Problem as follows: the problem is NP-complete, if k+l>2; the problem is polynomial otherwise. We consider the degree Δ constraint subproblem and completely classify the problem as follows: the problem is polynomial, for k⩽2 or Δ⩽3; the problem is NP-complete otherwise. In addition, we propose two optimization versions of graph sandwich problem for a property Π: MAX-Π-GSP and MIN-Π-GSP. We prove that MIN-(2,1)-GSP is a Max-SNP-hard problem, i.e., there is a positive constant ε, such that the existence of an ε-approximative algorithm for MIN-(2,1)-GSP implies P=NP

Even and odd pairs in comparability and in P4-comparability graphs

AbstractWe characterize even and odd pairs in comparability and in P4-comparability graphs. The characterizations lead to simple algorithms for deciding whether a given pair of vertices forms an even or odd pair in these classes of graphs. The complexities of the proposed algorithms are O(n + m) for comparability graphs and O(n2m) for P4-comparability graphs. The former represents an improvement over a recent algorithm of complexity O(nm)

Complexity dichotomy on partial grid recognition

Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even trivial. A natural step, outstanding thus far, was to provide a broad classification of graphs that make for polynomial or NP-complete instances. We provide such a classification based on the set of allowed vertex degrees in the input graphs, yielding a full dichotomy on the complexity of the problem. As byproducts, the previous NP-completeness result for binary trees was strengthened to strictly binary trees, and the three-dimensional version of the problem was for the first time proven to be NP-complete. Our results were made possible by introducing the concepts of consistent orientations and robust gadgets, and by showing how the former allows NP-completeness proofs by local replacement even in the absence of the latter

The polynomial dichotomy for three nonempty part sandwich problems

AbstractWe classify into polynomial time or NP-complete all three nonempty part sandwich problems. This solves the polynomial dichotomy into polynomial time and NP-complete for this class of graph partition problems

On the computational difficulty of the terminal connection problem

A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) ⊆ W ⊆ V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The Terminal connectio

The perfection and recognition of bull-reducible Berge graphs

The recently announced Strong Perfect Graph Theorem states that the class of perfect graphs coincides with the class of graphs containing no induced odd cycle of length at least 5 or the complement of such a cycle. A graph in this second class is called Berge. A bull is a graph with five vertices x, a, b, c, d and five edges xa, xb, ab, ad, bc. A graph is bull-reducible if no vertex is in two bulls. In this paper we give a simple proof that every bull-reducible Berge graph is perfect. Although this result follows directly from the Strong Perfect Graph Theorem, our proof leads to a recognition algorithm for this new class of perfect graphs whose complexity, O(n6), is much lower than that announced for perfect graphs

Finding

We study the concept of an H-partition of the vertex set of a graph G, which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H, with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, 4-part, external constraints only (no internal constraints), each part non-empty. We describe tools that yield for each problem considered in this paper a simple and low complexity polynomial-time algorithm

Split Clique Graph complexity

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C(G) the clique family of G. The clique graph of G, denoted by K(G), is the intersection graph of C(G). Say that G is a clique graph if there exists a graph H such that G=K(H). The clique graph recognition problem, a long-standing open question posed in 1971, asks whether a given graph is a clique graph and it was recently proved to be NP-complete even for a graph G with maximum degree 14 and maximum clique size 12. Hence, if P ≠ NP, the study of graph classes where the problem can be proved to be polynomial, or of more restricted graph classes where the problem remains NP-complete is justified. We present a proof that given a split graph G=(V,E) with partition (K,S) for V, where K is a complete set and S is a stable set, deciding whether there is a graph H such that G is the clique graph of H is NP-complete. As a byproduct, we prove that determining whether a given set family admits a spanning family satisfying the Helly property is NP-complete. Our result is optimum in the sense that each vertex of the independent set of our split instance has degree at most 3, whereas when each vertex of the independent set has degree at most 2 the problem is polynomial, since it is reduced to the problem of checking whether the clique family of the graph satisfies the Helly property. Additionally, we show three split graph subclasses for which the problem is polynomially solvable: the subclass where each vertex of S has a private neighbor, the subclass where |S|≤3, and the subclass where |K|≤4.Fil: Alcón, Liliana Graciela. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata; ArgentinaFil: Faria, Luerbio. Universidade do Estado de Rio do Janeiro; BrasilFil: De Figueiredo, Celina M.H.. Universidade Federal do Rio de Janeiro; BrasilFil: Gutierrez, Marisa. Universidad Nacional de La Plata; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

On Nordhaus–Gaddum type inequalities for the game chromatic and game coloring numbers

International audienceA seminal result by Nordhaus and Gaddum states that 2n≤χ(G)+χ(G¯)≤n+1 for every graph G of order n, where G¯ is the complement of G and χ is the chromatic number. We study similar inequalities for χg(G) and colg(G), which denote, respectively, the game chromatic number and the game coloring number of G. Those graph invariants give the score for, respectively, the coloring and marking games on G when both players use their best strategies