Papers by Marisa Gutierrez
arXiv (Cornell University), 2011
An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of... more An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. In this paper we characterize [h,2,1] graphs using chromatic number. We show that the problem of deciding whether a given VPT graph belongs to [h,2,1] is NP-complete, while the problem of deciding whether the graph belongs to [h,2,1]-[h-1,2,1] is NP-hard. Both problems remain hard even when restricted to $Split \cap VPT$. Additionally, we present a non-trivial subclass of $Split \cap VPT$ in which these problems are polynomial time solvable.
Lecture Notes in Computer Science, 2011
A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a m... more 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 longstanding 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 6 = 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| 6 3, and the subclass where |K | 6 4.
arXiv (Cornell University), Nov 16, 2012
Graph pebbling is a network optimization model for transporting discrete resources that are consu... more Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of two pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding if a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and that deciding if the pebbling number has a prescribed upper bound is Π P 2 -complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter three chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(n β ) time, where β = 2ω/(ω + 1) ∼ = 1.41 and ω ∼ = 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.
SSRN Electronic Journal
Graph convexity has been used as an important tool to better understand the structure of classes ... more Graph convexity has been used as an important tool to better understand the structure of classes of graphs. Many studies are devoted to determine if a graph equipped with a convexity is a convex geometry. In this work we survey results on characterizations of well-known classes of graphs via convex geometries. We also give some contributions to this subject.
arXiv (Cornell University), Apr 14, 2016
Graph pebbling is a network model for transporting discrete resources that are consumed in transi... more Graph pebbling is a network model for transporting discrete resources that are consumed in transit. Deciding whether a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and deciding whether the pebbling number has a prescribed upper bound is Π P 2complete. Recently we proved that the pebbling number of a split graph can be computed in polynomial time. This paper advances the program of finding other polynomial classes, moving away from the large tree width, small diameter case (such as split graphs) to small tree width, large diameter, continuing an investigation on the important subfamily of chordal graphs called k-trees. In particular, we provide a formula, that can be calculated in polynomial time, for the pebbling number of any semi-2-tree, falling shy of the result for the full class of 2-trees.
Discrete Mathematics
Graph convexity spaces have been studied in many contexts. In particular, some studies are devote... more Graph convexity spaces have been studied in many contexts. In particular, some studies are devoted to determine if a graph equipped with a convexity space is a convex geometry. It is well known that chordal and Ptolemaic graphs can be characterized as convex geometries with respect to the geodesic and monophonic convexities, respectively. Weak polarizable graphs, interval graphs, and proper interval graphs can also be characterized in this way. In this paper we introduce the notion of l k -convexity, a natural restriction of the monophonic convexity. Let G be a graph and k ≥ 2 an integer. A subset S ⊆ V (G) is l k -convex if and only if for any pair of vertices x, y of S, each induced path of length at most k connecting x and y is completely contained in the subgraph induced by S. The l k -convexity consists of all l k -convex subsets of G. In this work, we characterize l k -convex geometries (graphs that are convex geometries with respect to the l k -convexity) for k ∈ {2, 3}. We show that a graph G is an l 2 -convex geometry if and only if G is a chordal P 4 -free graph, and an l 3 -convex geometry if and only if G is a chordal graph with diameter at most three such that its induced gems satisfy a special "solving" property. As far as the authors know, the class of l 3 -convex geometries is the first example of a non-hereditary class of convex geometries.
Chordal graphs, which are intersection graph of subtrees of a tree, can be represented on trees. ... more Chordal graphs, which are intersection graph of subtrees of a tree, can be represented on trees. Some representation of a chordal graph often reduces the size of the data structure needed to store the graph, permitting the use of extremely efficient algorithms that take advantage of the compactness of the representation. An extended star graph is the intersection graph of a family of subtrees of a tree that has exactly one vertex of degree at least three. An asteroidal triple in a graph is a set of three non-adjacent vertices such that for any two of them there exists a path between them that does not intersect the neighborhood of the third. Several subclasses of chordal graphs (interval graphs, directed path graphs) have been characterized by forbidden asteroids. In this paper, we define, a subclass of chordal graphs, called extended star graphs, prove a characterization of this class by forbidden asteroids and show open problems.
Ars Comb., 2015
It will be proved that the problem of determining whether a set of vertices of a dually chordal g... more It will be proved that the problem of determining whether a set of vertices of a dually chordal graphs is the set of leaves of a tree compatible with it can be solved in polynomial time by establishing a connection with finding clique trees of chordal graphs with minimum number of leaves.
It is known that any vertex of a chordal graph has an eccentric vertex which is simplicial. Here ... more It is known that any vertex of a chordal graph has an eccentric vertex which is simplicial. Here we prove similar properties in related classes of graphs where the simplicial vertices will be replaced by other special types of vertices.
ArXiv, 2016
The edge intersection graph of a family of paths in host tree is called an $EPT$ graph. When the ... more The edge intersection graph of a family of paths in host tree is called an $EPT$ graph. When the host tree has maximum degree $h$, we say that $G$ belongs to the class $[h,2,2]$. If, in addition, the family of paths satisfies the Helly property, then $G \in$ Helly $[h,2,2]$. The time complexity of the recognition of the classes $[h,2,2]$ inside the class $EPT$ is open for every $h> 4$. Golumbic et al. wonder if the only obstructions for an $EPT$ graph belonging to $[h,2,2]$ are the chordless cycles $C_n$ for $n> h$. In the present paper, we give a negative answer to that question, we present a family of $EPT$ graphs which are forbidden induced subgraphs for the classes $[h,2,2]$. Using them we obtain a total characterization by induced forbidden subgraphs of the classes Helly $[h,2,2]$ for $h\geq 4$ inside the class $EPT$. As a byproduct, we prove that Helly $EPT$$\cap [h,2,2]=$ Helly $[h,2,2]$. We characterize Helly $[h,2,2]$ graphs by their atoms in the decomposition by cliq...
Order, 2020
A k-tree is either a complete graph on k vertices or a graph that contains a vertex whose neighbo... more A k-tree is either a complete graph on k vertices or a graph that contains a vertex whose neighborhood induces a complete graph on k vertices and whose removal results in a ktree. If the comparability graph of a poset P is a k-tree, we say that P is a k-tree poset. In the present work, we study and characterize by forbidden subposets the k-tree posets that admit a containment model mapping vertices into paths of a tree (CP T k-tree posets). Furthermore, we characterize the dually-CP T and strong-CP T k-tree posets and their comparability graphs. The characterizations lead to efficient recognition algorithms for the respective classes.
Electronic Notes in Theoretical Computer Science, 2019
A graph is CIS if every maximal clique interesects every maximal stable set. Currently, no good c... more A graph is CIS if every maximal clique interesects every maximal stable set. Currently, no good characterization or recognition algorithm for the CIS graphs is known. We characterize graphs in which every maximal matching saturates all vertices of degree at least two and use this result to give a structural, efficiently testable characterization of claw-free CIS graphs. We answer in the negative a question of Dobson, Hujdurović, Milanič, and Verret [Vertex-transitive CIS graphs, European J. Combin. 44 (2015) 87-98] asking whether the number of vertices of every CIS graph is bounded from above by the product of its clique and stability numbers. On the positive side, we show that the question of Dobson et al. has an affirmative answer in the case of claw-free graphs.
Discrete Mathematics, 2017
The edge-intersection graph of a family of paths on a host tree is called an EPT graph. When the ... more The edge-intersection graph of a family of paths on a host tree is called an EPT graph. When the tree has maximum degree h, we say that the graph is [h, 2, 2]. If, in addition, the family of paths satisfies the Helly property, then the graph is Helly [h, 2, 2]. In this paper, we present a family of EPT graphs called gates which are forbidden induced subgraphs for [h, 2, 2] graphs. Using these we characterize by forbidden induced subgraphs the Helly [h, 2, 2] graphs. As a byproduct we prove that in getting a Helly EPT -representation, it is not necessary to increase the maximum degree of the host tree. In addition, we give an efficient algorithm to recognize Helly [h, 2, 2] graphs based on their decomposition by maximal clique separators.
Discrete Mathematics, 2017
Graph pebbling is a network model for transporting discrete resources that are consumed in transi... more Graph pebbling is a network model for transporting discrete resources that are consumed in transit. Deciding whether a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and deciding whether the pebbling number has a prescribed upper bound is Π P 2complete. Recently we proved that the pebbling number of a split graph can be computed in polynomial time. This paper advances the program of finding other polynomial classes, moving away from the large tree width, small diameter case (such as split graphs) to small tree width, large diameter, continuing an investigation on the important subfamily of chordal graphs called k-trees. In particular, we provide a formula, that can be calculated in polynomial time, for the pebbling number of any semi-2-tree, falling shy of the result for the full class of 2-trees.
Discrete Applied Mathematics, 2016
In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-int... more In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NPcomplete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.
Discussiones Mathematicae Graph Theory, 2016
A graph is a path graph if there is a tree, called U V -model, whose vertices are the maximal cli... more A graph is a path graph if there is a tree, called U V -model, whose vertices are the maximal cliques of the graph and for each vertex x of the graph the set of maximal cliques that contains it induces a path in the tree. A graph is an interval graph if there is a U V -model that is a path, called an interval model. Gimbel [3] characterized those vertices in interval graphs for which there is some interval model where the interval corresponding to those vertices is an end interval. In this work, we give a characterization of those simplicial vertices x in path graphs for which there is some U V -model where the maximal clique containing x is a leaf in this U V -model.
Electronic Notes in Discrete Mathematics, 2015
Golumbic, Lipshteyn and Stern proved that every graph can be represented as the edge intersection... more Golumbic, Lipshteyn and Stern proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, B k -EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B 4 -EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B 3 -EPG, and that there exist circular-arc graphs which are not B 2 -EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define B k -EPR graphs, k ≥ 0, the same way as B k -EPG graphs. Circular-arc graphs are clearly B 4 -EPR graphs and we will show that there exist circular-arc graphs that are not B 3 -EPR graphs. We also show that normal circulararc graphs are B 2 -EPR graphs and that there exist normal circular-arc graphs that are not B 1 -EPR graphs. Finally, we characterize B 1 -EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.
Graphs and Combinatorics, 2015
A directed path graph is the intersection graph of a family of directed subpaths of a directed tr... more A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Clearly, rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted path graphs. With the purpose of proving knowledge in this direction, we show in this paper properties of directed path models that are not rootable for chordal graphs with any leafage and with leafage four. Therefore, we prove that for leafage four directed path graphs minimally non rooted path graphs has a unique asteroidal quadruple and can be characterized by the presence of certain type of asteroidal quadruples.
Discrete Applied Mathematics, 2016
A subtree of a directed path is a directed path.
European Journal of Combinatorics, 2015
A walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W... more A walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W , and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W . In the resulting interval convexity, a set S ⊂ V (G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.
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Papers by Marisa Gutierrez