Gutman and Degree Monophonic Index of Graphs

A graph with p points and q edges is denoted by G(p,q). An edge joining two non-adjacent points of a path P is called a chord of a path P. A path P is called monophonic if it is a chordless path. For any two points u and v in a connected graph G, the monophonic distance d (u, v) m from u to v is defined as the length of a longest u-v monophonic path in G. The Gutman monophonic index of a graph G is denoted by GutMP(G) and defined by GutMP(G)  d(u)d(v)d (u,v) m and degree monophonic index of G is denoted by DMP(G) and defined by DMP(G) d(u) d(v)d (u,v)    m . The methodology executed in this research paper is to determine the monophonic distance matrix of graphs under consideration. The entries of monophonic distance matrix are calculated by counting the number of edges in the u-v monophonic path. In this paper for some standard graphs, GutMP(G) and DMP(G) are studied which can be applied to derive quantitative structure- property or structure- activity relationships (QSPR / QSAR).

Extreme Monophonic Graphs and Extreme Geodesic Graphs

For a connected graph $G=(V,E)$ of order at least two, a chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path $P$ is called a monophonic path if it is a chordless path. A monophonic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a monophonic path joining some pair of vertices in $S$. The monophonic number of $G$ is the minimum cardinality of its monophonic sets and is denoted by $m(G)$. A geodetic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a geodesic joining some pair of vertices in $S$. The geodetic number of $G$ is the minimum cardinality of its geodetic sets and is denoted by $g(G)$. The number of extreme vertices in $G$ is its extreme order $ex(G)$. A graph $G$ is an extreme monophonic graph if $m(G)=ex(G)$ and an extreme geodesic graph if $g(G)=ex(G)$. Extreme monophonic graphs of order $p$ with monophonic number $p$ and $p-1$ are characterized. It is shown that every pair $a,b$ of integers with $0 \leq a \leq b$ is realized as the extreme order and monophonic number, respectively, of some graph. For positive integers $r,d$ and $k \geq 3$ with $r < d$, it is shown that there exists an extreme monophonic graph $G$ of monophonic radius $r$, monophonic diameter $d$, and monophonic number $k$. Also, we give a characterization result for a graph $G$ which is both extreme geodesic and extreme monophonic.

Equivalence between Hypergraph Convexities

Let G be a connected graph on V. A subset X of V is all-paths convex (or ap -convex) if X contains each vertex on every path joining two vertices in X and is monophonically convex (or m-convex) if X contains each vertex on every chordless path joining two vertices in X. First of all, we prove that ap -convexity and m-convexity coincide in G if and only if G is a tree. Next, in order to generalize this result to a connected hypergraph H, in addition to the hypergraph versions of ap -convexity and m-convexity, we consider canonical convexity (or c-convexity) and simple-path convexity (or sp -convexity) for which it is well known that m-convexity is finer than both c-convexity and sp -convexity and sp -convexity is finer than ap -convexity. After proving sp -convexity is coarser than c-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of γ-acyclic hypergraphs.

Induced Paths in Twin-Free Graphs

Let $G=(V,E)$ be a simple, undirected graph. Given an integer $r \geq 1$, we say that $G$ is $r$-twin-free (or $r$-identifiable) if the balls $B(v,r)$ for $v \in V$ are all different, where $B(v,r)$ denotes the set of all vertices which can be linked to $v$ by a path with at most $r$ edges. These graphs are precisely the ones which admit $r$-identifying codes. We show that if a graph $G$ is $r$-twin-free, then it contains a path on $2r+1$ vertices as an induced subgraph, i.e. a chordless path.

P_4-Colorings and P_4-Bipartite Graphs

International audience A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.

Linear time recognition of P4-indifference graphs

International audience A graph is a P4-indifference graph if it admits an ordering < on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has a

Linear Time Recognition of P4-Indifferent Graphs

<p>A simple graph is P4-indifferent if it admits a total order < on<br />its nodes such that every chordless path with nodes a, b, c, d and edges<br />ab, bc, cd has a < b < c < d or a > b > c > d. P4-indifferent graphs generalize<br /> indifferent graphs and are perfectly orderable. Recently, Hoang,<br />Maray and Noy gave a characterization of P4-indifferent graphs in<br />terms of forbidden induced subgraphs. We clarify their proof and describe<br /> a linear time algorithm to recognize P4-indifferent graphs. When<br />the input is a P4-indifferent graph, then the algorithm computes an order < as above.</p><p>Key words: P4-indifference, linear time, recognition, modular decomposition.</p><p> </p>