Sandwich and probe problems for excluding paths

The final publication is available at Elsevier via https://doi.org/10.1016/j.dam.2018.05.054 © 2018 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Let Pk denote an induced path on k vertices. For k ≥ 5, we show that the Pk-free sandwich problem, partitioned probe problem, and unpartitioned probe problem are NP-complete. For k ≤ 4, it is known that the Pk-free sandwich problem, partitioned probe problem, and unpartitioned probe problem are in P.The first author was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq Grant 303622/2011-3

The Sandwich Problem for Decompositions and Almost Monotone Properties

This is a post-peer-review, pre-copyedit version of an article published in Algorithmica. The final authenticated version is available online at: https://doi.org/10.1007/s00453-018-0409-6We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in C as an induced subgraph is almost monotone if C is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length ≡ j mod n is almost monotone if and only if j ≡ 2 mod n or n ≤ 2. Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.Maria Chudnovsky was supported by National Science Foundation Grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. Celina M. H. de Figueiredo was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq Grant 303622/2011-3

On Eggleton and Guy conjectured upper bound for the crossing number of the n-cube

. The crossing number (G) of a graph G is the smallest integer such that there is a drawing for G with (G) crossings of edges. Let Q n denote the n--dimensional cube. Eggleton and Guy conjectured in 1970 that (Q n ) 4 n 5 32 \Gamma 2 n\Gamma2 b n 2 +1 2 c. We exhibit a drawing for n = 6 with the same value of Eggleton and Guy's conjectured upper bound. We construct a family of drawings for the n--cubes, n 7, with number of crossings 165 1024 4 n \Gamma 2n 2 \Gamma11n+34 2 2 n\Gamma2 , establishing a new upper bound for (Q n ). Our family of drawings confirms Eggleton and Guy's conjectured upper bound when n = 7 and 8. In addition, our upper bound improves the upper bound (Q n ) 4 n 1 6 \Gamma2 n\Gamma3 n 2 \Gamma 2 n\Gamma4 3 + (\Gamma2) n 1 48 due to Madej. AMS Subject Classification: 05C10 Keywords: topological graph theory, crossing numbers 1 Introduction A simple drawing D(G) of a graph G is a drawing of G on the plane such that no edge crosses its..

On the Eggleton and Guy conjectured upper bound for the crossing number of the n-cube

Let Q n denote the n--dimensional cube. In this paper, we exhibit drawings for n = 6, 7 and 8. In these cases the drawings confirm Eggleton and Guy's conjectured upper bound for the crossing number of the n-cube

Total chromatic number of unichord-free graphs

14 p. : il.A unichord is an edge that is the unique chord of a cycle in a graph. The class C of unichordfree graphs — that is, graphs that do not contain, as an induced subgraph, a cycle with a unique chord — was recently studied by Trotignon and Vušković (2010) [24], who proved strong structure results for these graphs and used these results to solve the recognition and vertex-colouring problems. Machado et al. (2010) [18] determined the complexity of the edge-colouring problem in the class C and in the subclass C′ obtained from C by forbidding squares. In the present work, we prove that the total-colouring problem is NP-complete when restricted to graphs in C. For the subclass C′ , we establish the validity of the Total Colouring Conjecture by proving that every non-complete {square, unichord}-free graph of maximum degree at least 4 is Type 1

Total chromatic number of chordless graphs

11 p. : il.A graph G is said to be chordless if no cycle in G has a chord. Chordless graphs are exactly the graphs whose line graphs are wheelfree, which implies a connection between the study of the chromatic index of chordless graphs and the study of the chromatic number of wheel-free graphs. Recent works investigate the chromatic index of chordless graphs and prove that every chordless graph with maximum degree at least 3 is Class 1 [?, ?]. In the present work we investigate the total chromatic number of chordless graphs and prove that every chordless graph with maximum degree at least 3 is Type 1

On Eggleton And Guy's Conjectured Upper Bound For The Crossing Number Of The n-Cube

The crossing number u(G) of a graph G is the smallest integer such that there is a drawing for G with u(G) crossings of edges. Let Q_n denote the n-dimensional cube. Eggleton and Guy conjectured in 1970 that u(Q_n) \Xi . We exhibit a drawing for n = 6 with the same value of..

A decomposition for total-coloring partial-grids and list-total-coloring outerplanar graphs

9 p. : il.The total chromatic number χT (G) is the least numberof colors sufficient to color the elements (vertices and edges) of a graphG in such away that no incident or adjacent elements receive the same color. In the presentwork,we obtain two results on total-coloring. First, we extend the set of partial-grids classified with respect to the total-chromatic number, by proving that every 8-chordal partial-grid of maximum degree 3 has total chromatic number 4. Second, we prove a result on list-total-coloring biconnected outerplanar graphs. If for each element x of a biconnected outerplanar graph G there exists a set Lx of colors such that |Luw| = max{deg(u) + 1, deg(w) + 1} for each edge uw and |Lv| = 7−δdeg(v),3−2δdeg(v),2 (where δi ,j = 1 if i = j and δi ,j = 0 if i = j ) for each vertex v, then there is a total-coloring π of graph G such that π(x) ∈ Lx for each element x of G. The technique used in these two results is a decomposition by a cutset of two adjacent vertices, whose properties are discussed in the article