Computing K-terminal reliability of d-trapezoid graphs

Given a probabilistic graph, with reliable edges and unreliable vertices, $K$ -terminal reliability problem is to find the probability that a given subset $K$ of vertices remains connected. The problem is #P-complete for general graphs, even so for some special graphs such as chordal graphs, and comparability graphs. However, polynomial time algorithms to solve the problem have been designed for some special graphs such as interval graphs, permutation graphs, and $d$ -trapezoid graphs. Existing time complexity of the polynomial time algorithm for the $d$ -trapezoid graph is $O(n^{2d+1})$ , which is a very high degree polynomial. This makes it impractical to solve large problem instances using the existing algorithm. Here, we propose a novel technique and use it to design a simple linear-time algorithm to solve the problem on $d$ -trapezoid graphs. As the complexity of our algorithm is linear, it is no more a difficulty to solve large problem instances in small amount of time.

Treelike Comparability Graphs

The solution of many ventilation problems involving networks (such as finding routes of minimum length, construction of a network of sections, determination of the cross sections of the initial network with minimum transmission capacity, and estimation of the quality of the initial information obtained in the production of depression records) is much simplified if, instead of sets of all routes and cross sections of the original network, we calculate with onl~ a subset of independent routes and cross sections of the same network [i, 2]. Existing graphical methods of constructing (cross) sections of the original network can mainly be used only to find sections of simple networks [2]. There are no effective methods of constructing independent routes of the original network. By the methods given below, one can relatively easily construct all the independent routes and sections for an Oriented graph (network) G = (X, U, M) with any number of branches, where X = {x,, xl, .... xl, xj, .... Xn} is the set of vertices (nodes), U = {uij} is the set of branches (arcs), and M = l~} is the set of paths in the network G from the -'~ntry"x, to the "exit" x n.

On edge transitivity of directed graphs

The maximum independent set problem (MaxIS) on general graphs is known to be NP-hard to approximate within a factor of $n^{1-epsilon}$, for any $epsilon > 0$. However, there are many ``easy classes of graphs on which the problem can be solved in polynomial time. In this context, an interesting question is that of computing the maximum independent set in a graph that can be expressed as the union of a small number of graphs from an easy class. The MaxIS problem has been studied on unions of interval graphs and chordal graphs. We study the MaxIS problem on unions of perfect graphs (which generalize the above two classes). We present an $O(sqrt{n})$-approximation algorithm when the input graph is the union of two perfect graphs. We also show that the MaxIS problem on unions of two comparability graphs (a subclass of perfect graphs) cannot be approximated within any constant factor.

: This paper introduces a class of 1 player games of perfect information, which we call complementing games; the player is allowed moves which complement the value of successive plays. A complementing game is symmetric if all noncomplement moves are reversible (i.e., form a symmetric relation). These games are naturally related to a class of machines we call symmetric complementing machines. Symmetric nondeterministic machines were studied in (Lewis and Papadimitriou, 801): they are identical to our symmetric complementing machines with complement moves allowed only on termination. (A companion paper to appear will describe the computational complexity of symmetric complementing and alternating machines.) Of particular interest is the complexity class Sum *CSYMLOG, which contains the outcome problem of symmetric complementing games with constant complement bound with game positions encoded in log space, and next move relations computable in log space. We show that the decision problem for a restricted quantified Boolean logic Sum *QBF sum is complete in Sum *CSYMLOG. We also show that Sum *CSYMLOG contains many well- known and common combinatiorial problems: minimum spanning forests; k- connectivity and k-connected components, and also the recognition problems for many classes of graphs: planar graphs of valence 3; chordal graphs; comparability graphs; interval graphs; split graphs, and permutation graphs.

We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X,Y∈P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C,I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal.We observe that there are problems, for instance Densestk-Subgraph and b-Coloring, that are polynomial-time solvable on split graphs but become NP -hard on well-partitioned chordal graphs. On the other hand, we show that the Geodetic Set problem, known to be NP -hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs.We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles).

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log2ψ(G)⌉. In this article, we show that for an interval graph G ⌈log2ψ(G)⌉⩽cub(G)⩽⌈log2ψ(G)⌉+2. It is not clear whether the upper bound of ⌈log2ψ(G)⌉+2 is tight: till now we are unable to find any interval graph with cub(G)>⌈log2ψ(G)⌉. We also show that for an interval graph G, cub(G)⩽⌈log2α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ⌈log2α2⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)⩽cub(G). From the above result, it follows that for any graph G, cub(G)⩽box(G)⌈log2α⌉. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010

The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight.

On CCE graphs of doubly partial orders

CENTERS OF TRIANGULATED GRAPHS V. D. Chepoi Let G = (X, U) be an ordinary graph with arbitrarily (not necessarily finitely) many vertexes, any two of which are joined by some finite chain. We endow G with a standard metric d(x, y) equal to the number of edges in the chain of shortest length joining vertexes x, y. The eccentricity e(z) of a vertex z is defined as {d(z, v): v e X}. The radius r(G) is the least eccentricity of the vertexes, and the diameter d(G) is the largest eccentricity. The center C(G) of G is the subgraph generated by the set of vertexes with minimal eccentricity. It is well known [I, 2] that any graph G, even if it is not connected, is the center of some graph G', i.e., G = C(G'). At the same time, if one confines attention to special classes of graphs, their centers may have rather specific features. Thus, a well-known re- sult of Jordan [3] states that the center of any finite tree consists of one vertex or two adjacent vertexes; that is to say, the only two possibilities are the graphs K l and K 2 (where K n denotes the complete subgraph on n vertexes). The centers of maximal outerplanar graphs and 2-trees were described in [4, 5]. In this paper we characterize the centers of triangu- lated graphs. In this connection, we note that metric properties, including in particular properties of the centers, of triangulated graphs have been studied by various authors [6-9]. Our Theorems 1 and 2 were proved in [8], but they are established here in a more general form and the proofs are simpler. Recall [i0] that a graph G is said to be triangulated if, in any simple cycle F of length more than 3, there are two vertexes not adjacent in r but joined by an edge in G. Recall moreover that a clique of a graph G is any maximal complete subgraph (with re- spect to inclusion). The density of a graph G is the cardinality of the largest clique in G (if such a clique exists). Throughout this paper, the term triangulated graph will mean a triangulated graph of finite diameter without infinite complete subgraphs. Let CT denote the family of graphs which are centers of triangulated graphs. We shall need a number of additional concepts and definitions. Given sets M c X in a graph G and any number r ~ 0, we put where

A set of vertices D is a dominating set for a graph $G = (V,E)$ if every vertex not in D is adjacent to a vertex in D. A set of vertices is a total dominating set if every vertex in V is adjacent to a vertex in D. Cockayne, Goodman and Hedetniemi presented a linear time algorithm to determine minimum dominating sets for trees. Booth and Johnson established the NP-completeness of the problem for undirected path graphs. This paper presents a linear time algorithm to determine minimum total dominating sets of a tree and shows that for undirected path graphs the problem remains NP-complete.

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

If $K$ is a simplicial complex on $m$ vertices, the flagification of $K$ is the minimal flag complex $K^{f}$ on the same vertex set that contains $K$ . Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$ . This induces a sequence of maps of polyhedral products $(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$ . We show that $\unicode[STIX]{x1D6FA}f$ and $\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\text{}\underline{CY},\text{}\underline{Y})^{K}$ is a co- $H$ -space if and only if the 1-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and $f\circ g$ have right homotopy inverses in this case.

Finding a central vertex in an HHD-free graph