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Register a new userCharacterization and recognition of proper tagged probe interval graphs
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Shamik Ghosh Prof.
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Interval graphs were used in the study of genomics by the famous molecular biologist Benzer. Later on probe interval graphs were introduced by Zhang as a generalization of interval graphs for the study of cosmid contig mapping of DNA. A tagged probe interval graph (briefly, TPIG) is motivated by similar applications to genomics, where the set of vertices is partitioned into two sets, namely, probes and nonprobes and there is an interval on the real line corresponding to each vertex. The graph has an edge between two probe vertices if their corresponding intervals intersect, has an edge between a probe vertex and a nonprobe vertex if the interval corresponding to a nonprobe vertex contains at least one end point of the interval corresponding to a probe vertex and the set of non-probe vertices is an independent set. This class of graphs have been defined nearly two decades ago, but till today there is no known recognition algorithm for it. In this paper, we consider a natural subclass of TPIG, namely, the class of proper tagged probe interval graphs (in short PTPIG). We present characterization and a linear time recognition algorithm for PTPIG. To obtain this characterization theorem we introduce a new concept called canonical sequence for proper interval graphs, which, we belief, has an independent interest in the study of proper interval graphs. Also to obtain the recognition algorithm for PTPIG, we introduce and solve a variation of consecutive $1$s problem, namely, oriented consecutive $1$s problem and some variations of PQ-tree algorithm. We also discuss the interrelations between the classes of PTPIG and TPIG with probe interval graphs and probe proper interval graphs.
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The interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each intersecting pair of intervals. A probe interval graph is a variant that is motivated by an application to genomics, where the intervals are partitioned into two sets: probes and non-probes. The graph has an edge between two vertices if they intersect and at least one of them is a probe. We give a linear-time algorithm for determining whether a given graph and partition of vertices into probes and non-probes is a probe interval graph. If it is, we give a layout of intervals that proves this. We can also determine whether the layout of the intervals is uniquely constrained within the same time bound. As part of the algorithm, we solve the consecutive-ones probe matrix problem in linear time, develop algorithms for operating on PQ trees, and give results that relate PQ trees for different submatrices of a consecutive-ones matrix.
A proper edge-coloring of a graph $G$ with colors $1,ldots,t$ is called an emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $vin V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $mathfrak{N}_{c}$. For a graph $Gin mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_{c}(G)$ and $W_{c}(G)$, respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if $G$ is a triangle-free graph with at least two vertices and $Gin mathfrak{N}_{c}$, then $W_{c}(G)leq vert V(G)vert +Delta(G)-2$. We also obtain bounds on $w_{c}(G)$ and $W_{c}(G)$ for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.
An edge-coloring of a graph $G$ with consecutive integers $c_{1},ldots,c_{t}$ is called an emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $mathfrak{N}$. In 2004, Giaro and Kubale showed that if $G,Hin mathfrak{N}$, then the Cartesian product of these graphs belongs to $mathfrak{N}$. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the first author showed that if $G,Hin mathfrak{N}$ and $H$ is a regular graph, then $G[H]in mathfrak{N}$. In this paper, we prove that if $Gin mathfrak{N}$ and $H$ has an interval coloring of a special type, then $G[H]in mathfrak{N}$. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if $Gin mathfrak{N}$ and $T$ is a tree, then $G[T]in mathfrak{N}$.
A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total $t$-coloring of a graph $G$ is a total coloring of $G$ with colors $1,ldots,t$ such that all colors are used, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. In this paper we prove that all complete multipartite graphs with the same number of vertices in each part are interval total colorable. Moreover, we also give some bounds for the minimum and the maximum span in interval total colorings of these graphs. Next, we investigate interval total colorings of hypercubes $Q_{n}$. In particular, we prove that $Q_{n}$ ($ngeq 3$) has an interval total $t$-coloring if and only if $n+1leq tleq frac{(n+1)(n+2)}{2}$.
Let $G$ be a graph on $n$ vertices. For $iin {0,1}$ and a connected graph $G$, a spanning forest $F$ of $G$ is called an $i$-perfect forest if every tree in $F$ is an induced subgraph of $G$ and exactly $i$ vertices of $F$ have even degree (including zero). A $i$-perfect forest of $G$ is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. Moreover, we show that to decide whether $G$ has a 0-perfect forest with at least $|V(G)|/2+k$ edges, where $k$ is the parameter, is W[1]-hard. We also prove that for a prescribed edge $e$ of $G,$ it is NP-hard to obtain a 0-perfect forest containing $e,$ but one can decide if there existsa 0-perfect forest not containing $e$ in polynomial time. It is easy to see that every graph with odd number of vertices has a 1-perfect forest. It is not the case for proper 1-perfect forests. We give a characterization of when a connected graph has a proper 1-perfect forest.
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