Van Bang Le - Academia.edu (original) (raw)
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Graduate Center of the City University of New York
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Given a class of graphs, mathcalG\mathcal{G}mathcalG , a graph G is a probe graph of mathcalG\mathcal{G}mathcalG if its verti... more Given a class of graphs, mathcalG\mathcal{G}mathcalG , a graph G is a probe graph of mathcalG\mathcal{G}mathcalG if its vertices can be partitioned into two sets, ℙ (the probes) and ℕ (the nonprobes), where ℕ is an independent set, such that G can be embedded into a graph of mathcalG\mathcal{G}mathcalG by adding edges between certain nonprobes. In this paper we study the probe graphs of ptolemaic graphs when the partition of vertices is unknown. We present some characterizations of probe ptolemaic graphs and show that there exists a polynomial-time recognition algorithm for probe ptolemaic graphs.
Discrete Mathematics, 2012
ABSTRACT The class of intersection graphs of unit intervals of the real line whose ends may be op... more ABSTRACT The class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral.
Block graphs are graphs in which every block (biconnected component) is a clique. A graph G =(V, ... more Block graphs are graphs in which every block (biconnected component) is a clique. A graph G =(V, E)is said to be an (unpartitioned) probe block graph if there exist an independent set N ⊆V and some set E⊆N^2 such that the graph G=(V, E∪E') is a block graph; if such an independent set N is given, G is called a partitioned probe block graph. In this note we give good characterizations for probe block graphs, in both unpartitioned and partitioned cases. As a result, partitioned and unpartitioned probe block graphs can be recognized in linear time.
Given a class of graphs, mathcalG\mathcal{G}mathcalG , a graph G is a probe graph of mathcalG\mathcal{G}mathcalG if its verti... more Given a class of graphs, mathcalG\mathcal{G}mathcalG , a graph G is a probe graph of mathcalG\mathcal{G}mathcalG if its vertices can be partitioned into two sets, ℙ (the probes) and ℕ (the nonprobes), where ℕ is an independent set, such that G can be embedded into a graph of mathcalG\mathcal{G}mathcalG by adding edges between certain nonprobes. In this paper we study the probe graphs of ptolemaic graphs when the partition of vertices is unknown. We present some characterizations of probe ptolemaic graphs and show that there exists a polynomial-time recognition algorithm for probe ptolemaic graphs.
Discrete Mathematics, 2012
ABSTRACT The class of intersection graphs of unit intervals of the real line whose ends may be op... more ABSTRACT The class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral.
Block graphs are graphs in which every block (biconnected component) is a clique. A graph G =(V, ... more Block graphs are graphs in which every block (biconnected component) is a clique. A graph G =(V, E)is said to be an (unpartitioned) probe block graph if there exist an independent set N ⊆V and some set E⊆N^2 such that the graph G=(V, E∪E') is a block graph; if such an independent set N is given, G is called a partitioned probe block graph. In this note we give good characterizations for probe block graphs, in both unpartitioned and partitioned cases. As a result, partitioned and unpartitioned probe block graphs can be recognized in linear time.