The Minimum Reload s-t Path/Trail/Walk Problems

The carbonate-bearing metaperidotite from Sidironero Complex, north of the Xanthi town is composed primarily of olivine and orthopyroxene megacrysts and of Ti-clinohumite, tremolite, chlorite, dolomite, magnesite, talc, antigorite and spinel group minerals. The metaperidotite underwent a prograde HP metamorphism probably isofacial with the neighboring amphibolitized eclogites. Calculated P-T and P(T)-XCO2 phase diagram sections (pseudosections) for the bulk rock composition showed that XCO2 in the fluid phase was extremely low (≤0.008) at the first stages of the metamorphism and increased up to 0.022 at the peak P-T conditions ~1.5 GPa and 690 0C. The prograde metamorphism probably started from a hydrated and carbonated assemblage including talc+chlorite+magnesite+dolomite and proceeded with tremolite and antigorite formation before olivine growth, and orthopyroxene formation after olivine growth (Ol-1). Matrix dolomite, breakdown of chlorite (Chl-1) to Cr spinel+olivine and of Ti-clinohumite to olivine+Mg-ilmenite occurred during decompression. The P-T path is constrained by the absence of clinopyroxene in the metaperidotite.

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On paths, trails and closed trails in edge-colored graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.586 ◽

2012 ◽

Vol Vol. 14 no. 2 (Graph Theory) ◽

Author(s):

Laurent Gourvès ◽

Adria Lyra ◽

Carlos A. Martinhon ◽

Jérôme Monnot

Keyword(s):

Polynomial Time ◽

Optimal Solution ◽

Colored Graph ◽

Edge Disjoint ◽

Colored Graphs ◽

International Audience ◽

Np Complete ◽

T Path ◽

Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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